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Two random variables $X$ and $Y$ have variances and covariance that satisfy the following relationships:

$Var(X) = 2Var(Y)$.

$Cov(X,Y) = Var(Y).$

Let $A = X+2Y$ and $B=2X+Y$.

Calculate the correlation coefficient between $A$ and $B$.

I'm not sure how to go about solving this question, any help would be appreciated. Thank you.

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closed as off-topic by Jack, Did, JonMark Perry, José Carlos Santos, Claude Leibovici Jul 26 '17 at 6:45

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  • $\begingroup$ Where did you get stuck when applying the definition: $ \rho(AB)=\frac{E(AB)-E(A)E(B)}{\sqrt{Var(A)Var(B)}} $? $\endgroup$ – Jack Jul 25 '17 at 14:56
  • $\begingroup$ I added what I've written so far, when applying that definition do I just substitute the functions of X and Y for A and B? $\endgroup$ – Matthew Higgins Jul 25 '17 at 15:03
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Just use bilinearity of covariance: Let $\alpha, \beta$ be scalars, and $X,Y,Z$ be random variables. Then, $cov(\alpha X + \beta Y , Z) = \alpha cov(X,Z) + \beta cov(Y,Z)$. Also, $cov(X,Z) = cov(Z,X)$.

Note that $cov(A,B) = cov(X+2Y,2X+Y) = cov(X,2X+Y) + cov(2Y,2X+Y) = cov(X,2X)+cov(X,Y)+cov(2Y,2X)+cov(2Y,Y)=2 cov(X,X) + cov(X,Y) + 4 cov(Y,X) + 2 cov(Y,Y) = 2 var(X)+5cov(X,Y)+2 var(Y)$.

Similarly, $var(A)=var(X+2Y)= cov(X+2Y,X+2Y) = var(X)+4var(Y)+4 cov(X,Y)$ and $var(B) = cov(2X+Y,2X+Y) = 4 var(X) + var(Y) + 4 cov(X,Y)$.

Now, substitute these into the definition of the correlation coefficient $\frac{cov(A,B)}{\sqrt{var(A)var(B)}}$.

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  • $\begingroup$ Thank you, I didn't know about the bilinearity of covariance, this helped a lot. $\endgroup$ – Matthew Higgins Jul 25 '17 at 15:18

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