Finding correlation of two independent variables

Question

So I'm given $E(X), E(Y), V(X)$ and $V(Y)$ for two independent variables Y and X. I'm also given $\rho(X,Y)$. The math problem I'm working with then defines two new variables;

$U=X+Y$ and $W=2Y$

I'm then asked to find the correlation between U and W and given the hint to first find $V(U+W)$.

So I know that $Corr(U,W)=\frac{Cov(U,W)}{\sqrt{Var(U)Var(W)}}$.

But I don't understand how I'm supposed to use the hint and the value for $\rho(X,Y)$ to find the correlation. Really appreciate some help :)

Answer 1

Don't use the hint. It is not useful. Rather, you should first find $\mathsf{Var}(X+Y)$.

Likewise express all three factors in terms of known values. To get you started:

$$\begin{align}\mathsf{Var}(U)&=\mathsf{V}(X+Y)\\&=\ldots\\[2ex]\mathsf{Var}(W)&=\mathsf{Var}(2Y)\\&=4\,\mathsf{Var}(Y)\\[2ex]\mathsf{Cov}(U,W)&=\mathsf{Cov}(X+Y, 2Y)\\&=\ldots\end{align}$$

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Also remember that $\rho(X,Y)=\mathsf{Corr}(X,Y)$.