# Homework Help: Variance and Covariance

1. Mar 13, 2010

### CescGoal

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

1. The problem statement, all variables and given/known data
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) $$\rho$$XZ = $$\frac{Cov(X,Z)}{\sqrt{Var(X)*Var(Z)}}$$

= $$\frac{Cov(X,Z)}{\sqrt{9}*\sqrt{16}}$$
= $$\frac{Cov(X,Z)}{12}$$

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

2. Mar 13, 2010

### Dick

Re: Variance/Covariance

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

3. Mar 13, 2010

### CescGoal

Re: Variance/Covariance

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

4. Mar 13, 2010

### Dick

Re: Variance/Covariance

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.