What is the correlation of X and Z in terms of variance and covariance?

[CescGoal]

Very sorry that I've double posted but I realized i placed the original post in Precalculus.



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.

 

[Dick]

Cov(X,Z)=Cov(X,X-Y). Cov(X,X-Y)=Cov(X,X)-Cov(X,Y), right?

 

[CescGoal]

I didn't know about that law; thankyou very much.

 

[Dick]

I didn't know about that law; thankyou very much.

It's pretty obvious if you write out the definition of Cov. You should try and do that so you can see why it's true.