Download - Basic Prob Concepts
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Probability distributionfunctions
Normal distribution
Lognormal distribution
Mean, median and mode Tails
Extreme value distributions
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Normal (Gaussian)distribution
Probability density function (PD)
!"at does #gure tell about t"e cumulative
distribution function ($D)%
1 1( ) exp
22
xf x
=
( ) ( ) ( )
x
F x P X x f t dt= < =
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More on t"e normaldistribution
Normal distribution is denoted , &it" t"e s'uaregiving t"e variance
f * is normal,Y=aX+b is also normal !"at&ould be t"e mean and standard deviation of Y%
+imilarly, if * and are normal variables, anylinear combination, aX+bYis also normal
$an often use any function of a normal randomvariables by using a linear Taylor ex-ansion
Exam-le.X=N(10,0.52)and Y=X2.T"en Y
N(100,102)
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Estimating mean and standard
deviation
Given a sam-le from a normally distributed variable,t"e sam-le mean is t"e best linear unbiasedestimator (0L1E) of t"e true mean
or t"e variance t"e e'uation gives t"e best
unbiased estimator, but t"e s'uare root is not anunbiased estimate of t"e standard deviation
or exam-le, for a sam-le of 2 from a standardnormal distribution, t"e standard deviation &ill beestimated on average as 345 (&it" standarddeviation of 365)
( )2
2
1 1
1 1
1
n n
i i
i i
x x x x
n n
= =
=
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Lognormal distribution
f ln(*) "as normal distribution * "aslognormal distribution T"at is, if * isnormally distributed ex-(*) is
lognormally distributed Notation.
PD
Mean and variance
( )2
2
ln1( ) exp
22
xf x
x
=
( ) ( ) ( )2 2
2 2 2exp / 2 , 1X X
Var X e e += + = =
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Mean, mode and median
Mode ("ig"est -oint) 7 Median (238 of sam-les)
igure for 73
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Lig"t and "eavy tails
Normal distribution "as lig"t tail9 52 sigmais e'uivalent to 65e:; failure or defect
-robability
Lognormal can "ave "eavy tail
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itting distribution to data
1sually #t $D to minimi test)
Generated ?3 -oints from N(3,12).
Normal #t N(3.48,0.932
) Lognormal lnN(@?5,3?;)
Almost same mean and
standard deviation
1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
CDF
experimental
lognormal
normal
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Extreme value distributions
No matter &"at distribution you sam-le from, t"emean of t"e sam-le tends to be normally distributedas sam-le si
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Maximum of normalsam-les
!it" normal distribution, maximum of sam-le is morenarro&ly distributed t"an original distribution
-1 0 1 2 3 4 5 60
1000
2000
3000
4000
5000
6000
7000
8000
Max of @3standard
normalsam-les @25mean, 324standarddeviation
1 1.5 2 2.5 3 3.5 4 4.5 5 5.50
1000
2000
3000
4000
5000
6000
7000
8000
9000
Max of @33standard normalsam-les ?23mean, 356standarddeviation
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Gumbel distribution
Mean, median, mode and variance
-5 -4 -3 -2 -1 0 10
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0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
fitted ev1
-max10 data
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -10
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fitted ev1
-max100 data
( )
1
exp , exp( )
z zx
PDF z e z CDF e
= = =
2
2
ln(ln(2)) mode=
Euler-Mascheroni constant 0.5772
Mean median
Variance
= + =
=
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!eibull distribution Probability distribution
ts log "as Gumbel dist 1sed to describe distribution of strengt" or fatigue life in brittle
materials f it describes time to failure, t"en
BC@ indicates t"at failure rate decreases &it" time, B7@ indicates constant rate,
B@ indicates increasing rate $an add 6rd-arameter by re-lacing x by x:c
-8 -6 -4 -2 0 2 40
0.1
0.2
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1
log weibull
ev1 fit
( )1
/
( ! , ) 0, 0, 0
kk
xk x
f x k e x k
= > >
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Exercises
ind "o& many sam-les of normally distributed numbersyou need in order to estimate t"e mean and standard
deviation &it" an error t"at &ill be less t"an @38 of t"etrue standard deviation most of t"e time
0ot" t"e lognormal and !eibull distributions are used tomodel strengt" ind "o& closely you can a--roximate
data generated from a standard lognormal distribution by#tting it &it" !eibull
TaBe t"e introduction and -reamble of t"e 1+ Declaration
of nde-endence, and #t t"e distribution of &ord lengt"susing t"e =:+ criterion !"at distribution #ts best%
$om-are t"e gra-"s of t"e $Ds $om-are to a morecontem-orary text