Correlation of U=Z+X and V=Z+Y

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Homework Statement



##X##, ##Y## and ##Z## are uncorrelated random variables with variances [itex]\sigma^{2}_{X}[/itex], [itex]\sigma^{2}_{Y}[/itex] and [itex]\sigma^{2}_{Z}[/itex], respectively. ##U=Z+X## and ##V=Z+Y##. Find [itex]\rho_{UV}[/itex].

Homework Equations



[itex]\rho_{UV}=\frac{Cov(U,V)}{\sqrt{Var(U)Var(V)}}[/itex]

The Attempt at a Solution



Since X, Y and Z are uncorrelated, Cov(X,Y)=Cov(X,Z)=Cov(Y,Z)=0. So,

##Cov(U,V)=Cov(Z+X,Z+Y)=Var(Z)=##

The variances are

##Var(U)=Var(Z+X)=Var(Z)+Var(X)##
##Var(V)=Var(Z+Y)=Var(Z)+Var(Y)##

Putting it all together gives

[itex]\rho_{UV}=\frac{Var(Z)}{\sqrt{[Var(Z)+Var(X)][Var(Z)+Var(Y)]}}[/itex]

This seems correct. Could it be simplified further?
 
Last edited:

Answers and Replies

  • #3
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Thanks.
 

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