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

[cborse]

Homework Statement

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

Homework Equations

$\rho_{UV}=\frac{Cov(U,V)}{\sqrt{Var(U)Var(V)}}$

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

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

This seems correct. Could it be simplified further?

 

[mfb]

I don't see how it could be simplified.

 

[cborse]

Thanks.