distribution functions introduction section 0 lecture 1 slide 1 lecture 3 slide 1 introduction to...
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DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 1
Lecture 3 Slide 1
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Lecture 3
Distribution Functions
References Taylor Ch 5 (and Chs 10 and 11 for Reference) Taylor Ch 6 and 7
Also refer to ldquoGlossary of Important Terms in Error Analysisrdquo ldquoProbability Cheat Sheetrdquo
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 2
Lecture 3 Slide 2
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Distribution Functions
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 3
Lecture 3 Slide 3
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Practical Methods to Calculate Mean and St Deviation
We need to develop a good way to tally display and think about a collection of repeated measurements of the same quantity
Here is where we are headedbull Develop the notion of a probability distribution function a distribution to
describe the probable outcomes of a measurementbull Define what a distribution function is and its propertiesbull Look at the properties of the most common distribution function the Gaussian
distribution for purely random eventsbull Introduce other probability distribution functions
We will develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random measurements)bull Schwartz inequality (ie the uncertainty principle) (next lecture)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorem
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 4
Lecture 3 Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
Flip Penny 50 times and record the resultsFlip a penny 50 times and record the results
Grab a partner and a set of instructions and complete the exercise
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry (kurtosis)
What is the probability of rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 2
Lecture 3 Slide 2
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Distribution Functions
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 3
Lecture 3 Slide 3
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Practical Methods to Calculate Mean and St Deviation
We need to develop a good way to tally display and think about a collection of repeated measurements of the same quantity
Here is where we are headedbull Develop the notion of a probability distribution function a distribution to
describe the probable outcomes of a measurementbull Define what a distribution function is and its propertiesbull Look at the properties of the most common distribution function the Gaussian
distribution for purely random eventsbull Introduce other probability distribution functions
We will develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random measurements)bull Schwartz inequality (ie the uncertainty principle) (next lecture)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorem
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 4
Lecture 3 Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
Flip Penny 50 times and record the resultsFlip a penny 50 times and record the results
Grab a partner and a set of instructions and complete the exercise
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry (kurtosis)
What is the probability of rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 3
Lecture 3 Slide 3
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Practical Methods to Calculate Mean and St Deviation
We need to develop a good way to tally display and think about a collection of repeated measurements of the same quantity
Here is where we are headedbull Develop the notion of a probability distribution function a distribution to
describe the probable outcomes of a measurementbull Define what a distribution function is and its propertiesbull Look at the properties of the most common distribution function the Gaussian
distribution for purely random eventsbull Introduce other probability distribution functions
We will develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random measurements)bull Schwartz inequality (ie the uncertainty principle) (next lecture)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorem
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 4
Lecture 3 Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
Flip Penny 50 times and record the resultsFlip a penny 50 times and record the results
Grab a partner and a set of instructions and complete the exercise
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry (kurtosis)
What is the probability of rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 4
Lecture 3 Slide 4
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
Flip Penny 50 times and record the resultsFlip a penny 50 times and record the results
Grab a partner and a set of instructions and complete the exercise
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry (kurtosis)
What is the probability of rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 5
Lecture 3 Slide 5
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Flip a penny 50 times and record the results
Two Practical Exercises in Probabilities
What is the asymmetry of the results
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry (kurtosis)
What is the probability of rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 6
Lecture 3 Slide 6
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Flip a penny 50 times and record the results
What is the asymmetry of the results
4 asymmetry asymmetry
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry (kurtosis)
What is the probability of rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 7
Lecture 3 Slide 7
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities
Roll a pair of dice 50 times and record the results
What is the mean value The standard deviation
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry (kurtosis)
What is the probability of rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 8
Lecture 3 Slide 8
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Practical Exercises in Probabilities Roll a pair of dice 50 times and record the results
What is the mean value
The standard deviation
What is the asymmetry (kurtosis)
What is the probability of rolling a 4
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 9
Lecture 3 Slide 9
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Discrete Distribution Functions
A data set to play with Written in terms of ldquooccurrencerdquo F
The mean value In terms of fractional expectations
Fractional expectations
Normalization condition
Mean value
(This is just a weighted sum)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 10
Lecture 3 Slide 10
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limit of Discrete Distribution Functions
ldquoNormalizingrdquo data sets
Binned data sets
fk equiv fractional occurrence
∆k equiv bin width
Mean value
Normalization
Expected value 119875119903119900119887ሺ4ሻ= 1198654119873= 1198914
119883= 119865119896119896 119909119896 = (119891119896119896 Δ119896)119909119896
1 = σ (119891119896119896 Δ119896)119909119896σ Δ119896119896
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 11
Lecture 3 Slide 11
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Limits of Distribution FunctionsConsider the limiting distribution function as N ȸ and ∆k0
Larger data sets
Mathcad Games
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 12
Lecture 3 Slide 12
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Continuous Distribution Functions
Central (mean) Value Width of Distribution
Meaning of Distribution Interval
Normalization of Distribution
Thus
and by extension
= fraction of measurements that fall within altxltb
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 13
Lecture 3 Slide 13
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
The first moment for a probability distribution function is
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119891(119909)+infinminusinfin 119889119909
For a general distribution function
119909ҧequiv 119909ۃ =ۄ 119891119894119903119904119905 119898119900119898119890119899119905 = 119909 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis
0 (for a centered distribution)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 14
Lecture 3 Slide 14
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Moments of Distribution Functions
Generalizing the nth moment is
119909119899 equiv 119909119899ۃ =ۄ 119899119905ℎ 119898119900119898119890119899119905 = 119909119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = 119909119899 119891(119909)+infinminusinfin 119889119909
Oth moment equiv N 2nd moment equiv ۃሺ119909minus 119909ҧሻ2 rarrۄ 1199092ۃ ۄ1st moment equiv 119909ҧ 3rd moment equiv kurtosis The nth moment about the mean is 120583119899 equiv minusሺ119909ۃ 119909ҧሻ119899 =ۄ 119899119905ℎ 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ119899 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ119899 119891(119909)+infinminusinfin 119889119909
The standard deviation (or second moment about the mean) is
1205901199092 equiv 1205832 equiv minusሺ119909ۃ 119909ҧሻ2 =ۄ 2119899119889 119898119900119898119890119899119905 119886119887119900119906119905 119905ℎ119890 119898119890119886119899
= ሺ119909minus119909ҧሻ2 119892(119909)+infinminusinfin 119889119909 119892(119909)+infinminusinfin 119889119909 = ሺ119909minus 119909ҧሻ2 119891(119909)+infinminusinfin 119889119909
0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 15
Lecture 3 Slide 15
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesHarmonic Oscillator Example from Mechanics
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 16
Lecture 3 Slide 16
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesBoltzmann Distribution Example from Kinetic Theory
Expected Values
The expectation value of a function Ɍ(x) is
Ɍሺxሻۃ equivۄ Ɍሺxሻ119892ሺ119909ሻ+infinminusinfin 119889119909 119892ሺ119909ሻ+infinminusinfin 119889119909
= Ɍ(x) 119891(119909)+infinminusinfin 119889119909
The Boltzmann distribution function for velocities of particles as a function of temperature T is
119875ሺ119907119879ሻ= 4120587 ൬ 1198722120587 119896119861119879൰3 2Τ 1199072119890119909119901ۏێێێ121198721199072ۍ 12119896119861119879൙
ےۑۑۑې
Then
vۃ =ۄ 119907 119875(119907)+infinminusinfin 119889119907 = ቂ8 119896119861119879 120587 119872ൗ ቃ1 2Τ
v2ۃ =ۄ v2 119875(119907)+infinminusinfin 119889119907 = ቂ3 119896119861119879 119872ൗቃ1 2Τ
implies ۃKE =ۄ 12119872 v2ۃ =ۄ 32119896119861119879
vpeak=ඨ2 119896119861119879 119872ൗ ൨1 2Τ = 2 3ൗ൩1 2Τ
v2ۃ ۄ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 17
Lecture 3 Slide 17
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFermi-Dirac Distribution Example from Kinetic Theory
For a system of identical fermions the average number of fermions in a single-particle state is given by the FermindashDirac (FndashD) distribution
where kB is Boltzmanns constant T is the absolute temperature is the energy of the single-particle state and μ is the total chemical potential
Since the FndashD distribution was derived using the Pauli exclusion principle which allows at most one electron to occupy each possible state a result is that
When a quasi-continuum of energies has an associated density of states (ie the number of states per unit energy range per unit volume) the average number of fermions per unit energy range per unit volume is
where is called the Fermi function
so that
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 18
Lecture 3 Slide 18
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Example of Continuous Distribution Functions and Expectation ValuesFinite Square Well Example from Quantum Mechanics
Expectation Values The expectation value of a QM operator O(x) is ۃOሺxሻ equivۄ Ψlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψሺxሻ119889119909 ΨlowastሺxሻΨሺxሻ+infinminusinfin 119889119909
For a finite square well of width L Ψnሺxሻ= ට2 Lൗ sinቂn π xL ቃ
Ψnlowastሺxሻ|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119874ሺ119909ሻ+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1
119909ۃ =ۄ Ψnlowastሺxሻ|x|Ψnሺxሻۃ equivۄ Ψnlowastሺxሻ119909+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 1198712
119901ۃ =ۄ |Ψnlowastሺxሻۃ ℏi partpartx|Ψnሺxሻ equivۄ Ψnlowastሺxሻℏi partpartx+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 0
119864119899ۃ =ۄ |Ψnlowastሺxሻۃ iℏ partpartt|Ψnሺxሻ equivۄ Ψnlowastሺxሻiℏ partpartt+infinminusinfin Ψnሺxሻ119889119909 ΨnlowastሺxሻΨnሺxሻ+infinminusinfin 119889119909 = 11989921205872ℏ22 119898 1198712
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 19
Lecture 3 Slide 19
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Distribution Functions
Available on web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 20
Lecture 3 Slide 20
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function
References Taylor Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 21
Lecture 3 Slide 21
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Integrals
Factorial Approximations 119899 asympሺ2120587119899ሻ1 2Τ 119899119899 119890119909119901ቂminus119899+ 112 119899 + 119874ቀ
11198992ቁቃ 119897119900119892ሺ119899ሻasymp 12 119897119900119892ሺ2120587ሻ+ቀ119899+ 12ቁlogሺ119899ሻminus 119899+ 112 119899 + 119874ቀ
11198992ቁ
119897119900119892ሺ119899ሻasymp 119899logሺ119899ሻminus 119899 (for all terms decreasing faster than linearly with n)
Gaussian Integrals 119868119898 = 2 119909119898 exp (minus1199092)infin0 119889119909 mgt-1 119868119898 = 2 119910119899 exp (minus119910)infin0 119889119910equiv Γ(119899+ 1) 1199092 equiv 119910 2 119889119909= 1199101 2Τ 119889119910 119899 equiv 12 (119898minus 1) 1198680 = Γቀ119899 = 12ቁ= ξ120587 m=0 119899 = minus12 1198682 119896 = Γቀ119896+ 12ቁ=൫119896minus 1 2ൗ൯൫119896minus 3 2ൗ൯൫3 2ൗ൯൫1 2ൗ൯ ξ120587 even m m=2 kgt0 119899 = 119896minus 12 1198682 119896+1 = Γሺ119896+ 1ሻ= 119896 odd m m=2 k+1gt0 119899 = 119896 ge 0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 22
Lecture 3 Slide 22
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Gaussian Distribution Function
Width of Distribution(standard deviation)
Center of Distribution(mean)Distribution
Function
Independent Variable
Gaussian Distribution Function
NormalizationConstant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 23
Lecture 3 Slide 23
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Effects of Increasing N on Gaussian Distribution Functions
Consider the limiting distribution function as N infin and dx0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 24
Lecture 3 Slide 24
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Defining the Gaussian distribution function in Mathcad
I suggest you ldquoinvestigaterdquo these with the Mathcad sheet on the web site
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 25
Lecture 3 Slide 25
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Using Mathcad to define other common distribution functions
Consider the limiting distribution function as N ȸ and ∆k0
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 26
Lecture 3 Slide 26
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 27
Lecture 3 Slide 27
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is not symmetric about X
Example Cauchy Distribution
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 28
Lecture 3 Slide 28
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 29
Lecture 3 Slide 29
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
When is mean x not Xbest
Answer When the distribution is has more than one peak
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 30
Lecture 3 Slide 30
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
The Gaussian Distribution Function and Its Relation to
Errors
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 31
Lecture 3 Slide 31
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
A Review of Probabilities in Combination
Probability of a data set of N like measurements (x1x2hellipxN)
P (x1x2hellipxN) = P(x)1P(x2)hellipP(xN)
1 head AND 1 Four P(H4) = P(H) P(4)
1 head OR 1 Four P(H4) = P(H) + P(4) - P(H and 4)(true for a ldquonon-mutually exclusiverdquo events)
NOT 1 SixP(NOT 6) = 1 - P(6)
1 Six OR 1 Four P(64) = P(6) + P(4)(true for a ldquomutually exclusiverdquo single role)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 32
Lecture 3 Slide 32
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The Gaussian Distribution Function and Its Relation to Errors
We will use the Gaussian distribution as applied to random variables to develop the mathematical basis forbull Meanbull Standard deviationbull Standard deviation of the mean (SDOM)bull Moments and expectation valuesbull Error propagation formulasbull Addition of errors in quadrature (for independent and random
measurements)bull Numerical values for confidence limits (t-test)bull Principle of maximal likelihoodbull Central limit theorembull Weighted distributions and Chi squaredbull Schwartz inequality (ie the uncertainty principle) (next lecture)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 33
Lecture 3 Slide 33
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider the Gaussian distribution function
Use the normalization condition to evaluate the normalization constant (see Taylor p 132)
The mean Ẋ is the first moment of the Gaussian distribution function (see Taylor p 134)
The standard deviation σx is the standard deviation of the mean of the Gaussian distribution function (see Taylor p 143)
Gaussian Distribution Moments
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 34
Lecture 3 Slide 34
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Standard Deviation of Gaussian Distribution
Area under curve (probability that ndashσltxlt+σ) is 68
5 or ~2σ ldquoSignificantrdquo
1 or ~3σ ldquoHighly Significantrdquo
1σ ldquoWithin errorsrdquo
5 ppm or ~5σ ldquoValid for HEPrdquo
See Sec 106 Testing of Hypotheses
More complete Table in App A and B
Ah thatrsquos highly significant
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 35
Lecture 3 Slide 35
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
Complementary Error Function (probability that ndashxlt-tσ AND xgt+tσ )Prob(x outside tσ) = 1 - Prob(x within tσ)
More complete Table in App A and B
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 36
Lecture 3 Slide 36
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Useful Points on Gaussian Distribution
Full Width at Half MaximumFWHM
(See Prob 512)
Points of InflectionOccur at plusmnσ
(See Prob 513)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 37
Lecture 3 Slide 37
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Analysis and Gaussian Distribution
Adding a Constant Multiplying by a Constant
XX+A XBX and σB σ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 38
Lecture 3 Slide 38
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Sum of Two Variables
Consider the derived quantityZ=X + Y (with X=0 and Y=0)
Error Propagation Addition
Error in ZMultiple two probabilities
119875119903119900119887ሺ119909119910ሻ= 119875119903119900119887ሺ119909ሻ∙119875119903119900119887ሺ119909ሻprop 119890119909119901minus12ቆ11990921205901199092+ 11991021205901199102ቇ൨prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯minus1199112ቇ൨
prop 119890119909119901minus12ቆ (119909+119910)2൫1205901199092+1205901199102൯ቇ൨ 119890119909119901ቂminus11991122ቃ
ICBST (Eq 553)
X+YZ and σx2 + σy
2 σz2
(addition in quadrature for random independent variables)
Integrates to radic2π
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 39
Lecture 3 Slide 39
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for error propagation see [Taylor Secs 35 and 39]
Uncertainty as a function of one variable [Taylor Sec 35]
1 Consider a graphical method of estimating error a) Consider an arbitaray function q(x) b) Plot q(x) vs x c) On the graph label
(1) qbest = q(xbest) (2) qhi = q(xbest + δx) (3) qlow = q(xbest- δx)
d) Making a linear approximation
x
qqxslopeqq
x
qqxslopeqq
bestbestlow
bestbesthi
e) Therefore x
x
Note the absolute value
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 40
Lecture 3 Slide 40
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Error Propagation
General formula for uncertainty of a function of one variable x
x
[Taylor Eq 323]
Can you now derive for specific rules of error propagation 1 Addition and Subtraction [Taylor p 49] 2 Multiplication and Division [Taylor p 51] 3 Multiplication by a constant (exact number) [Taylor p 54] 4 Exponentiation (powers) [Taylor p 56]
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 41
Lecture 3 Slide 41
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
General Formula for Multiple VariablesUncertainty of a function of multiple variables [Taylor Sec 311]
1 It can easily (no really) be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
2 More correctly it can be shown that (see Taylor Sec 311) for a function of several variables
)(
zz
qy
y
qx
x
qzyxq [Taylor Eq 347]
where the equals sign represents an upper bound as discussed above 3 For a function of several independent and random variables
)(222
zz
qy
y
qx
x
qzyxq [Taylor Eq 348]
Again the proof is left for Ch 5
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 42
Lecture 3 Slide 42
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Product of Two Variables
Consider the arbitrary derived quantityq(xy) of two independent random variables x and y
Expand q(xy) in a Taylor series about the expected values of x and y (ie at points near X and Y)
Error Propagation General Case
119902ሺ119909119910ሻ= 119902ሺ119883119884ሻ+൬120597119902120597119909൰ฬ119883ሺ119909minus 119883ሻ+൬
120597119902120597119910൰ฬ119884(119910minus 119884)
Fixed shifts peak of distribution
Fixed Distribution centered at X with width σX
120575119902ሺ119909119910ሻ= 120590119902 = ඨ119902ሺ119883119884ሻ+ ൬120597119902120597119909൰ฬ119883120590119909൨2 +ቈ൬120597119902120597119910൰ฬ1198841205901199102
0
How do we determine the error of a derived quantity Z(XYhellip) from errors in XYhellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 43
Lecture 3 Slide 43
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
SDOM of Gaussian Distribution
Standard Deviation of the Mean
The SDOM decreases as the square root of the number of measurements
That is the relative width σẊ of the distribution gets narrower as more measurements are made
Each measurement has similar σXi= σẊ
and similar partial derivatives
Thushellip
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 44
Lecture 3 Slide 44
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Two Key Theorems from Probability
Central Limit Theorem
For random independent measurements (each with a well-define expectation value and well-defined variance) the arithmetic mean (average) will be approximately normally distributed
Principle of Maximum Likelihood
Given the N observed measurements x1 x2hellipxN the best estimates for Ẋ and σ are those values for which the observed x1 x2hellipxN are most likely
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 45
Lecture 3 Slide 45
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Mean of Gaussian Distribution as ldquoBest Estimaterdquo
Principle of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set
Each randomly distributed with
x1 x2 x3 hellipxN
To find the most likely value of the mean (the best estimate of ẋ) find X that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 46
Lecture 3 Slide 46
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Uncertainty of ldquoBest Estimatesrdquo of Gaussian DistributionPrinciple of Maximum Likelihood
Best Estimate of X is from maximum probability or minimum summation
Consider a data set x1 x2 x3 hellipxN
To find the most likely value of the standard deviation (the best estimate of the width of the x distribution) find σx that yields the highest probability for the data set
The combined probability for the full data set is the product
Minimize Sum
Solve for derivative wrst X set to 0
Best estimate of X
Best Estimate of σ is from maximum probability or minimum summation
Minimize Sum
Best estimate of σ
Solve for derivative wrst σ set to 0
See Prob 526
prop 1120590119890minus(1199091minus119883)2 2120590Τ times 1120590119890minus(1199092minus119883)2 2120590Τ times helliptimes 1120590119890minus(119909119873minus119883)2 2120590Τ = 1120590119873119890σminus(119909119894minus119883)2 2120590Τ
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 47
Lecture 3 Slide 47
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Combining Data Sets
Weighted Averages
References Taylor Ch 7
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 48
Lecture 3 Slide 48
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages
Question How can we properly combine two or more separate independent measurements of the same randomly distributed quantity to determine a best combined value with uncertainty
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 49
Lecture 3 Slide 49
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Consider two measurements of the same quantity described by a random Gaussian distribution ltx1gt x1 and ltx2gt x2 The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equiv ቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
Assume negligible systematic errors
Note χ2 or Chi squared is the sum of the squares of the deviations from the mean divided by the corresponding uncertainty
Such methods are called ldquoMethods of Least Squaresrdquo They follow directly from the Principle of Maximum Likelihood
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 50
Lecture 3 Slide 50
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
The probability of measuring two such measurements is 119875119903119900119887119909ሺ11990911199092ሻ= 119875119903119900119887119909ሺ1199091ሻ 119875119903119900119887119909ሺ1199092ሻ = 112059011205902 119890minus12059422 119908ℎ119890119903119890 1205942 equivቈ
ሺ1199091 minus 119883ሻ1205901 2 +ቈሺ1199092 minus 119883ሻ1205902 2
To find the best value for χ find the maximum Prob or minimum χ2
Weighted Averages
This leads to 119909119882_119886119907119892 = 11990811199091+119908211990921199081+1199082 = σ119908119894 119909119894σ119908119894 119908ℎ119890119903119890 119908119894 = 1 ሺ120590119894ሻ2ൗ
Best Estimate of χ is from maximum probibility or minimum summation
Minimize Sum Solve for best estimate of χSolve for derivative wrst χ set to 0
Note If w1=w2 we recover the standard result Xwavg= (12) (x1+x2)
Finally the width of a weighted average distribution is 120590119908119894119890119892ℎ119905119890119889 119886119907119892 = 1σ 119908119894119894
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 51
Lecture 3 Slide 51
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Weighted Averages on SteroidsA very powerful method for combining data from different sources with different methods and uncertainties (or indeed data of related measured and calculated quantities) is Kalman filtering
The Kalman filter also known as linear quadratic estimation (LQE) is an algorithm that uses a series of measurements observed over time containing noise (random variations) and other inaccuracies and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone
The Kalman filter keeps track of the estimated state of the system and the variance or uncertainty of the estimate The estimate is updated using a state transition model and measurements xk|k-1 denotes the estimate of the systems state at time step k before the k th measurement yk has been taken into account Pk|k-1 is the corresponding uncertainty --Wikipedia 2013
Ludger Scherliess of USU Physics is a world expert at using Kalman filtering for the assimilation of satellite and ground-based data and the USU GAMES model to predict space weather
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 52
Lecture 3 Slide 52
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Rejecting Data
Chauvenetrsquos Criterion
References Taylor Ch 6
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 53
Lecture 3 Slide 53
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never bull Whenever it makes things look betterbull Chauvenetrsquos criterion provides a (quantitative) compromise
Rejecting Data
What is a good criteria for rejecting data
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 54
Lecture 3 Slide 54
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Never
Rejecting Data
Zallenrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 55
Lecture 3 Slide 55
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Question When is it ldquoreasonablerdquo to discard a seemingly ldquounreasonablerdquo data point from a set of randomly distributed measurements
bull Whenever it makes things look better
Rejecting Data
Disneyrsquos Criterion
Disneyrsquos First Law
Wishing will make it so
Disneyrsquos Second Law
Dreams are more colorful than reality
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 56
Lecture 3 Slide 56
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Data may be rejected if the expected number of measurements at least as deviant as the suspect measurement is less than 50
Consider a set of N measurements of a single quantity x1x2 helliphellipxN Calculate ltxgt and σx and then determine the fractional deviations from the mean of all the points
For the suspect point(s) xsuspect find the probability of such a point occurring in N measurements n = (expected number as deviant as xsuspect) = N Prob(outside xsuspectσx)
Rejecting Data
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 57
Lecture 3 Slide 57
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Error Function of Gaussian Distribution
Area under curve (probability that ndashtσltxlt+tσ) is given in Table at right
Probable Error (probability that ndash067σltxlt+067σ) is 50
Error Function (probability that ndashtσltxlt+tσ )
Error Function Tabulated values-see App A
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 58
Lecture 3 Slide 58
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Criterion
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 59
Lecture 3 Slide 59
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 1
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 60
Lecture 3 Slide 60
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos Details (1)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 61
Lecture 3 Slide 61
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (2)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
- Slide 54
- Slide 55
- Slide 56
- Slide 57
- Slide 58
- Slide 59
- Slide 60
- Slide 61
- Slide 62
- Slide 63
- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 62
Lecture 3 Slide 62
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashDetails (3)
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Intermediate Lab PHYS 3870 (3)
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
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- Intermediate Lab PHYS 3870 (5)
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- Intermediate Lab PHYS 3870 (6)
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- Intermediate Lab PHYS 3870 (7)
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DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 63
Lecture 3 Slide 63
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Chauvenetrsquos CriterionmdashExample 2
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
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- Intermediate Lab PHYS 3870 (3)
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- Intermediate Lab PHYS 3870 (4)
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- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
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- Intermediate Lab PHYS 3870 (5)
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- Intermediate Lab PHYS 3870 (6)
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- Intermediate Lab PHYS 3870 (7)
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DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 64
Lecture 3 Slide 64
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Intermediate Lab PHYS 3870
Summary of Probability Theory
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
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- Intermediate Lab PHYS 3870 (3)
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- Intermediate Lab PHYS 3870 (4)
- Slide 31
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- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
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- Intermediate Lab PHYS 3870 (5)
- Slide 48
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- Intermediate Lab PHYS 3870 (6)
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- Intermediate Lab PHYS 3870 (7)
- Slide 65
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-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 65
Lecture 3 Slide 65
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-I
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
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- Intermediate Lab PHYS 3870 (3)
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- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
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- Intermediate Lab PHYS 3870 (5)
- Slide 48
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- Intermediate Lab PHYS 3870 (6)
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- Intermediate Lab PHYS 3870 (7)
- Slide 65
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- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 66
Lecture 3 Slide 66
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-II
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
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- Intermediate Lab PHYS 3870 (3)
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- Intermediate Lab PHYS 3870 (4)
- Slide 31
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- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
- Slide 49
- Slide 50
- Slide 51
- Intermediate Lab PHYS 3870 (6)
- Slide 53
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- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 67
Lecture 3 Slide 67
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-III
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
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- Intermediate Lab PHYS 3870 (3)
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- Intermediate Lab PHYS 3870 (4)
- Slide 31
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- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
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- Intermediate Lab PHYS 3870 (5)
- Slide 48
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- Intermediate Lab PHYS 3870 (6)
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- Intermediate Lab PHYS 3870 (7)
- Slide 65
- Slide 66
- Slide 67
- Slide 68
-
DISTRIBUTION FUNCTIONS
Introduction Section 0 Lecture 1 Slide 68
Lecture 3 Slide 68
INTRODUCTION TO Modern Physics PHYX 2710
Fall 2004
Intermediate 3870
Fall 2013
Summary of Probability Theory-IV
- Intermediate Lab PHYS 3870
- Intermediate Lab PHYS 3870 (2)
- Slide 3
- Slide 4
- Slide 5
- Slide 6
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- Intermediate Lab PHYS 3870 (3)
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- Intermediate Lab PHYS 3870 (4)
- Slide 31
- Slide 32
- Slide 33
- Slide 34
- Slide 35
- Slide 36
- Slide 37
- Slide 38
- General Formula for Error Propagation
- General Formula for Error Propagation (2)
- General Formula for Multiple Variables
- Slide 42
- Slide 43
- Slide 44
- Slide 45
- Slide 46
- Intermediate Lab PHYS 3870 (5)
- Slide 48
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- Intermediate Lab PHYS 3870 (6)
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- Intermediate Lab PHYS 3870 (7)
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