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Coefficient of correlation between Z1 and Z2

Raphaël Steenbergen, 15-01-2014

Many students did come with questions how to determine the coefficient of correlation or the covariance of two limit state functions.

Below you find the elaboration that can be applied to whatever expression for Z1 and Z2

We however use the example of the exam of April 2013

By definition it holds:

ρ (Z1 , Z2 )=cov (Z1 , Z2 )σZ1σZ2

σ Z1σZ2 can be determined very easily or is even given in the question.

Problem is to determine cov (Z1 , Z2 ). By definition it holds:

The elaboration is done below step by step for question 2b of the exam April 2013. The steps can be applied to different expressions of Z1 and Z2.

Given:

cov (Zc , Z s )=cov (1− hhs− QQc ,1− hhs )μ (Z c)=1−

μ (h )hc

−μ (Q )Qc

μ(Zs)=1−μ(h)hs

Solution:

We rewrite this expression:

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We make groups of variables in combination with each time its mean value:

We now apply the multiplication:

We recognize the expressions for the covariance and the variance:

We elaborate this taking deterministic parameters outside:

We know that and we also know that Q and h are independent, so :

We finally obtain: