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5. Combination of random variables

• Understand why we need bottoms-up approach for reliability analysis

• Learn how to compute the probability density function, mean value and standard deviation of functions of random variables. Also learn how to approximate the mean value and standard deviation of functions of random variables.

• We will assume static reliability models for the rest of the course.

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Bottoms-up approach for reliability analysis

Select primitive random variables

Probability distributions of primitive random variables

Probability calculus

Reliability or failure probability

Data and judgment

Relation between

performance and

primitive random variables

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Why bottoms-up approach for reliability analysis

• Sometimes we do not have enough failure data to estimate reliability of a system. Examples: buildings, bridges, nuclear power plants, offshore platforms, ships

• Solution: Bottom up approach for reliability assessment: start with the probability distributions of the primitive (generic random variables), derive probability distribution of performance variables (e.g. failure time).

• Advantages: – Estimate probability distribution of input random variables (e.g., yield

stress of steel, wind speed). Use the same probability distribution of the generic random variables in many different problems.

– Identify and reduce important sources of uncertainty and variability.

4

Transformation of random variables

• y=g(x)

• Objective: given probability distribution of X, and function g(.), derive probability distribution of Y.

5

Transformation of random variables

X

Y

x x+Dx

y

y+Dy

y=g(x)

)(gx where)()(

)( 1- y

dxxdgxf

yf XY

One-to-one transformation

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General transformationmultiple-valued inverse function

...)()(

)(

)()(

yg2x

X

yg1x

XY

11 dxdyxf

dxdyxf

yf

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Functions of many variables

X1

X2

Y1

Y2

Ax Ay

J

xxf

A

A

xxfyyf XX

x

y

XXYY

),(),(),( 2121

212121

21

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Expectation (mean value) and variance

• In many problems it is impractical to estimate probability density functions so we work with mean values (expectations) and variances

• Expectation– E(aX)=aE(X)– E(X+Y)=E(X)+E(Y)– If X, Y independent, then E(XY)=E(X)E(Y)

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Variance

YXYXYX

X

XaX

XaX

XEXE

a

2

)]([)(

222

222

22

222

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Covariance

• Covariance measures the degree to which two variables tend to increase to decrease together

X

Y Negative covariance

X

Y Positive covariance

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Correlation coefficient

• Correlation coefficient, : covariance normalized by the product of standard deviations

• Ranges from –1 to +1

• Uncorrelated variables: correlation coefficient=0

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Relation between correlation and statistical dependence

• If X, Y independent then they are uncorrelated

• If X, Y are uncorrelated, then they may be dependent or independent

Uncorrelatedvariables

Independentvariables

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Variance of uncorrelated variables

222YXYX

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Chebyshev’s inequality

• Upper bound of probability of a random variable deviating more than k standard deviations from its mean value

• P(|Y-E(Y)|k)1/k2

• Upper bound is to large to be useful

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k P(|Y-E(Y)|k)1/k2

from Chebyshev’sinequality

P(|Y-E(Y)|k)Y normal

1 1 0.3172 0.25 0.0463 0.11 0.0034 0.06 6*10E56 0.03 0

10 0.01 0

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Approximations for mean and variance of a function of random variables

• Function of one variable: g(X)

• E(g(X))=g(E(X))

• Standard deviation of g(X)=[dg(X)/dX]standard deviation of X

• Derivative of g(X) is evaluated at the mean value of X

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Approximations for mean and variance of a function of random variables

• Function of many variables: g(X1,…,Xn)

• E(g(X1,…,Xn))=g(E(X1),…, E(Xn))

• Variance of g(X) = [dg(X1,…,Xn)/dXi]2×variance of Xi+ 2 [dg(X1,…,Xn)/dXi] × [dg(X1,…,Xn)/dXj]

×covariance of Xi,Xj

• Derivatives of g(X) are evaluated at the mean value of X

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When are the above approximations good?

• When the standard deviations of the independent variables are small compared to their average values

• Function g(X) is mildly nonlinear i.e. the derivatives do not change substantially when the independent variables change