# Difficult problem

• Nov 6th 2008, 07:37 PM
wolverine21
Difficult problem
2. Let X1 and X2 be independent random variables, with means μ1 and μ2 and variances σ1^2 and σ2^2, respectively. Find the Covariance of Y1 = X1 and Y2 = X1 – X2.

So for this one is the covariance of y 1 just the standard formula of covariance without any manipulation? And how about y2?
• Nov 7th 2008, 11:31 PM
mr fantastic
Quote:

Originally Posted by wolverine21
2. Let X1 and X2 be independent random variables, with means μ1 and μ2 and variances σ1^2 and σ2^2, respectively. Find the Covariance of Y1 = X1[snip]

Cov(X1, X1) = Var(X1)
• Nov 7th 2008, 11:35 PM
CaptainBlack
Quote:

Originally Posted by wolverine21
2. Let X1 and X2 be independent random variables, with means μ1 and μ2 and variances σ1^2 and σ2^2, respectively. Find the Covariance of Y1 = X1 and Y2 = X1 – X2.

So for this one is the covariance of y 1 just the standard formula of covariance without any manipulation? And how about y2?

Because $\displaystyle X_1$ and $\displaystyle X_2$ their joint pdf is the product of their individial pdf's.

$\displaystyle E( (Y_1-\bar{Y_1})(Y_2-\bar{Y_2})=\iint (x_1-\bar{x_1})(x_1-x_2-\bar{x_1}+\bar{x_2}) p(x_1)p(x_2) dx_1 dx_2$

Which you should be able to complete

CB
• Nov 7th 2008, 11:36 PM
CaptainBlack
Quote:

Originally Posted by mr fantastic
Cov(X1, X1) = Var(X1)

Your snip is in the wrong place, the covariance of $\displaystyle Y_1$ and $\displaystyle Y_2$ is required.

CB
• Nov 7th 2008, 11:52 PM
mr fantastic
Quote:

Originally Posted by CaptainBlack
Your snip is in the wrong place, the covariance of $\displaystyle Y_1$ and $\displaystyle Y_2$ is required.

CB

(Doh)
• Nov 10th 2008, 06:37 PM
wolverine21
Quote:

Originally Posted by CaptainBlack
Because $\displaystyle X_1$ and $\displaystyle X_2$ their joint pdf is the product of their individial pdf's.

$\displaystyle E( (Y_1-\bar{Y_1})(Y_2-\bar{Y_2})=\iint (x_1-\bar{x_1})(x_1-x_2-\bar{x_1}+\bar{x_2}) p(x_1)p(x_2) dx_1 dx_2$

Which you should be able to complete

CB

Wait so is this joint pdf I find the final answer?
• Nov 10th 2008, 10:50 PM
CaptainBlack
Quote:

Originally Posted by wolverine21
Wait so is this joint pdf I find the final answer?

No, you need to complete the double integral, you don't have to know the actual pdf's for X_1 and X_1 it is sufficient to know their means and variances and that they are independant (so Cov(X_1,X_2)=0 amoung other things).

CB