# Finding correlation of two independent variables

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 :)

## 1 Answer

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}

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

• So, I have calculated both $Var(U)$ and $Var(W)$. I'm not sure how I should solve $Cov(U,V)$ from the formulas in my book, but I found this formula online; $Cov(aX+bY,cX+dY)=acVar(X)+(ad+bc)Cov(X,Y)+bdVar(Y)$. I suppose I can use this equation? Then $Cov(X+Y, 2Y)=2Cov(X,Y)+2Var(Y)$. And I find $Cov(X,Y)$ by taking $Corr(X,Y)*\sqrt{Var(X)*Var(Y)}$ Commented Sep 28, 2021 at 12:38
• Indeed. That is the way. PS: The equation is the Bilinearity of Covariance. @Mathomat55 Commented Sep 28, 2021 at 13:59
• You may also use the definition of covariance;:\small\begin{align}\mathsf{Cov}(X+Y, 2Y) &= \mathsf E\bigl((X+Y)(2Y)\bigr)-\mathsf E(X+Y)~\mathsf E(2Y)\\&=2\bigl(\mathsf E(XY)-\mathsf E(X)\,\mathsf E(Y)+\mathsf E(Y^2)-(\mathsf E(Y))^2\bigr)\\&=2\bigl(\mathsf{Cov}(X,Y)+\mathsf{Var}(Y)\bigr)\end{align} Commented Sep 28, 2021 at 14:06
• I see! Thank you so much for your help :) Commented Sep 28, 2021 at 15:44