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1. Homework Statement

Question

Let X and Y be independent random variables with variances 9 and 7 respectively and let

Z = X - Y

a) What is the value of Cov(X,Z)

b) What is the value of the correlation coefficient of X and Z?

I've been stuck on this one question for 2-3 hours; its ridiculous, I know. Here's my terrible try.

3. The Attempt at a Solution

a)

Var(X) = 9

Var(Y) = 7

Var(X-Y) = Var(X) + Var(Y) = Var(Z)

Therefore, Var(Z) = 7 + 9 =16

Cov(X,Z) = E[XZ] - E[X]E[Z]

and b) [tex]\rho[/tex]

_{XZ}= [tex]\frac{Cov(X,Z)}{\sqrt{Var(X)*Var(Z)}}[/tex]

= [tex]\frac{Cov(X,Z)}{\sqrt{9}*\sqrt{16}}[/tex]

= [tex]\frac{Cov(X,Z)}{12}[/tex]

Since last topic, I've realized that Cov(X,Y) = 0 due to independency. But I don't know how to use it.