basic prob concepts

7/25/2019 Basic Prob Concepts

1/13

Probability distributionfunctions

Normal distribution

Lognormal distribution

Mean, median and mode Tails

Extreme value distributions


2/13

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= < =


3/13

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 += + = =


6/13

Mean, mode and median

Mode ("ig"est -oint) 7 Median (238 of sam-les)

igure for 73


7/13

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


8/13

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


9/13

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


10/13

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

0.1

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

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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

= + =

=


12/13

!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

0.3

0.4

0.5

0.6

0.7

0.8

0.9

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

basic prob concepts

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