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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 :)

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

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    $\begingroup$ 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 $Corr(X,Y)*\sqrt{Var(X)*Var(Y)}$ $\endgroup$
    – Mathomat55
    Commented Sep 28, 2021 at 12:38
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    $\begingroup$ Indeed. That is the way. PS: The equation is the Bilinearity of Covariance. @Mathomat55 $\endgroup$ Commented Sep 28, 2021 at 13:59
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    $\begingroup$ You may also use the definition of covariance;:$$\small\begin{align}\mathsf{Cov}(X+Y, 2Y) &= \mathsf E\bigl((X+Y)(2Y)\bigr)-\mathsf E(X+Y)~\mathsf E(2Y)\\&=2\bigl(\mathsf E(XY)-\mathsf E(X)\,\mathsf E(Y)+\mathsf E(Y^2)-(\mathsf E(Y))^2\bigr)\\&=2\bigl(\mathsf{Cov}(X,Y)+\mathsf{Var}(Y)\bigr)\end{align}$$ $\endgroup$ Commented Sep 28, 2021 at 14:06
  • $\begingroup$ I see! Thank you so much for your help :) $\endgroup$
    – Mathomat55
    Commented Sep 28, 2021 at 15:44

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