Reliability analysis in Random variables engineering ... · Reliability analysis in engineering...

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Reliability analysis in engineering applications

Random variables and standard distributions

Structural Engineering - Lund University 1

Random variables and standard distributions

Random variablesA random variable is a function X(v) defined on a sample space Ώ. The random variable maps the

event v into R1.


Ω

)(vX v

1R

The random variable is generally described by it’s cumulative

distribution function, FX

Random variables

The cumulative distribution function, FX(x) defines the probability P that the random variable X is less

or equal to a certain value x.


( ) ( )xXPxXPxFX >−=≤= 1)(

Random variablesFor a distribution function, FX(x) for a R.V. X, the

following must be valid:

1. FX(x) → 0 then x → –∞0 6

0,8

1


2. FX(x) → 1 then x → ∞

3. If x increase, FX(x) increase (growing function)

This implies that if a≤b is:

P(a<X ≤b)=FX(b)-FX(a)

0

0,2

0,4

0,6

-3 -2 -1 0 1 2 3 4 5 6 7

xF X

(x)

Random variablesAnother way to describe a R.V. is by it’s

probability density function, fX(x).

0,3

0,4


0

0,1

0,2

-3 -2 -1 0 1 2 3 4 5 6 7

x

f X(x

)

fx describes how the probability density is distributed.

The area under fx is always 1.

Random variablesThere are two types of distributions:


Continuous

Discrete

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Random variablesA continuous R.V. has a continuous distribution function and

can take an infinite number of values.

0 6

0,8

1

0,6

0,8

1

fX(x)FX(x)


0

0,2

0,4

0,6

0 1 2 3 4 5

x

F X(x

)

0

0,2

0,4

,

0 1 2 3 4 5

x

f X(x

)

dxxFdxf X

X))(()( =

dxxfbXaPaFbFb

aXX ∫=≤<=− )()()()(

a b ba

Random variables

0,6

0,8

1

F X(x

)

0,2

0,3

f X(x

)

A discrete R.V. can only take a finite number of values.

fX(x)FX(x)


0

0,2

0,4

0 2 4 6 8

x

F

0

0,1

-2 0 2 4 6 8

x

∑ ≤=

kj xX jpkF )()(⎩⎨⎧

−−=

=annars )1()(

0k då )0()(

kFkFF

kpXX

Xx

Random variables

Mean value

0 8

1

Median1

Location parameters


0

0,2

0,4

0,6

0,8

0 1 2 3 4 5

x

f X(x

)

0

0,2

0,4

0,6

0,8

0 1 2 3 4 5

x

f X(x

) 50%

Center of mass 50 % of the area on each side of the median

(For a symmetric dist. the location parameters are the same.)

Random variables

∑ ⋅=k

X kpkXE )()(

Mean value or expected value

Discrete R.V.


Continuous R.V. ∫∞

∞−

⋅= dxxfxXE X )()(

The first moment, center of gravity

Random variablesScale parameters

fX


x

Random variables

∑ −=k

X kpmxXV )()()( 2

Variance

Discrete R.V.


∫∞

∞−

−= dxxfmxXV X )()()( 2

The variance is also called the second moment. Many standard distributions is described by their first two moments, e.g. the

normal distribution.

Continuous R.V.

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Random variables

The standard deviation D(X) is defined as:

)()( XVXD =


)()(cov

XEXD

=

The coefficient of variation, cov is the ratio between the standard devation and the mean value:

)()(

Standard probability distributions

q

Let’s assume that we are interested in determineing the failure probability for a steel beam in bending.


fy, WL

q

Limit state function: fyW-qL2/2=0

The normal distributionThe normal distribution,

N(μ,σ)

0,8

1

0 3

0,4

FX fX


0

0,2

0,4

0,6

-3 -2 -1 0 1 2 3 4 5 6 7

x

F X(x

)

0

0,1

0,2

0,3

-3 -2 -1 0 1 2 3 4 5 6 7

x

f X(x

)

The standard normal distribution, N(0,1)

The normal distribution

•Old distribution (the 19th century)

•Symmetric distribution


•Useful distribution to model e.g. permanent loads model uncertainties related to loads, DAF and

dimensions

•Good mathematic characteristics

The normal distributionExample:

If X is a normal distributed R.V. with mean value and standard deviation equal to 6 and 2 respectively. What

is the probability that x is larger than 4 and less or


is the probability that x is larger than 4 and less or equal than 7?

P(4 < x ≤ 7)

The log-normal distribution

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FX fX

If the logarithm of a R.V. X is normal distributed then X is log-normal distributed.


0

0,2

0,4

0,6

0,8

0 1 2 3 4 5

x

F X(x

)

0

0,2

0,4

0,6

0,8

0 1 2 3 4 5

x

f X(x

)

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The log-normal distributionParameters

),()ln( lnln σμNYX ∈=

ParametersMean value and


⎟⎟⎠

⎞⎜⎜⎝

⎛+=

2exp)(

2ln

lnσμYE

)))(()(ln(5,0))(ln(2 2ln YEYVYE +−=μ

)))((ln(2 lnln μσ −= YE))exp()2)(exp(2exp()( 2

ln2lnln σσμ −=YV

standard deviation

The log-normal distribution•Asymmetric

•Can't take negative values

•Useful distribution to model e.g. strength


Useful distribution to model e.g. strength parameters, model uncertainties related to

resistance and dimensions.

•Good mathematical features

The uniform distribution

fX

1/(b-a)

FX

1

U(a,b)


bax

ba x

Takes values between a and b with the same probability


Meeting between two vehicles

Useful distribution in simulations.


Bridge


3500

4000

V1+V2 2000

2500

Meeting at support

Meeting mid span


0

500

1000

1500

2000

2500

3000

0 10 20 30 40 50 60

Location of V1 [m]

Mom

ent [

kNm

]

V1

V2

V1 V2

0

500

1000

1500

0 10 20 30 40 50 60

Location of V1 [m]

Mom

ent [

kNm

]

V1+V2

V1

V2

The Poisson distribution

0,6

0,8

1

F X(x

)

Po(λ)

0,2

0,3

f X(x

)

FX(x) fX(x)


0

0,2

0,4

0 2 4 6 8

x

Discrete distribution

0

0,1

-2 0 2 4 6 8

x

The parameter, λ shall be interpreted as intensity.

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The Poisson distribution

The distribution describes the probability for X number events to come true.

X can be anything, e.g. the number of l h ll d i d h b f


telephone calls during a day or the number of meetings during a year between two heavy

vehicles.

The sum of two Poisson distributed R.V. with mean values λ1 and λ2 is also Poisson

distributed with mean value λ1 + λ2.

The exponential distribution

0,8

1

Exp(m)

FX(x) fX(x)

0,8

1


0

0,2

0,4

0,6

0 1 2 3 4 5

x

F X(x

)

0

0,2

0,4

0,6

0 1 2 3 4 5

x

f X(x

)

The exponential distributionOften useful to describe remaining lifetime of

different components, e.g. for how long time will a bulb work? Another example: What is the time

interval between to heavy vehicles?


The exponential distribution is also very useful to model extreme

values.

The exponential and the poission distinction

l3

l2

l1


e3e2e1

d2 d1t=0 t=

l: is uniformly dist. When in time or space do the events occur.

e: Is Poisson dist., number of events

d: exp. dist: time interval between two events

Fitting values to standard distributions

betafit - Beta parameter estimation.binofit - Binomial parameter estimation.dfittool - Distribution fitting tool.

Matlab ”help stats”


gevfit - Extreme value parameter estimation.expfit - Exponential parameter estimation.gamfit - Gamma parameter estimation.gevfit - Generalized extreme value parameter estimation.gpfit - Generalized Pareto parameter estimation.lognfit - Lognormal parameter estimation.nbinfit - Negative binomial parameter estimation.normfit - Normal parameter estimation.poissfit - Poisson parameter estimation.raylfit - Rayleigh parameter estimation.unifit - Uniform parameter estimation.wblfit - Weibull parameter estimation.

Simulations of R.V. Matlab ”help stats”

betarnd - Beta random numbers.binornd - Binomial random numbers.chi2rnd - Chi square random numbers.evrnd - Extreme value random numbers.exprnd - Exponential random numbers.


p pfrnd - F random numbers.gamrnd - Gamma random numbers.geornd - Geometric random numbers.gevrnd - Generalized extreme value random numbers.gprnd - Generalized Pareto inverse random numbers.lognrnd - Lognormal random numbers.ncx2rnd - Noncentral Chi-square random numbers.normrnd - Normal (Gaussian) random numbers.poissrnd - Poisson random numbers.unidrnd - Discrete uniform random numbers.unifrnd - Uniform random numbers.wblrnd - Weibull random numbers.

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Example: Determine the distribution for the sum you get when you throw 5 dices and determine the probability that

the result is less than 21?>> help unidrnd

UNIDRND Random arrays from the discrete uniform distribution


UNIDRND Random arrays from the discrete uniform distribution.R = UNIDRND(N) returns an array of random numbers chosen uniformly

from the set {1, 2, 3, ... ,N}. The size of R is the size of N.

R = UNIDRND(N,MM,NN,...) or R = UNIDRND(N,[MM,NN,...]) returns anMM-by-NN-by-... array.

>> unidrnd(6,1,5)3 6 5 2 6

>> sum(unidrnd(6,1,5))22

Example:

close all;clear all;

for i =1:10000b(i)=sum(unidrnd(6,1,5));

end 0 4

0.5

0.6

0.7

0.8

0.9

1

F(x)

Empirical and Normal estimated cdf


wnormfit(b)

5 10 15 20 25 300

0.1

0.2

0.3

0.4

xμ=17,6σ=3,8

normcdf(20,17.6,3.8)Ans = 0.2638

Goodness of fit

0,6

0,7

0,8

0,9

1Test data

Fitted distribution

0,2

0,3Test data

Fitted distribution


0

0,1

0,2

0,3

0,4

0,5

5 10 15

x

F X(x

)

0,0

0,1

5 7 9 11 13 15

x

f X(x

)

Goodness of fitvisual

600

700

QQ-plot


0

100

200

300

400

500

0 100 200 300 400 500 600 700

Empirical quantiles

Theo

retic

al q

uant

iles

Goodness of fitVisual Distribution

paper

0.9970.999

Normal Probability Plot


5 10 15 20 25

0.0010.0030.01 0.02 0.05 0.10

0.25

0.50

0.75

0.90 0.95 0.98 0.99

Data

Pro

babi

lity

Goodness of fitNumerical

•Mean square error

•Kolmogorov Smirnov


Kolmogorov Smirnov

•Andersson-Darlingtest

•Ҳ2-test

Reliability analysis in Random variables engineering ... · Reliability analysis in engineering...

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