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## Homework Statement

1. Show that [tex]X[/tex] is independent of [tex] Y- \alpha X [/tex]

where [tex]|\alpha| < 1 [/tex]

2. Find the joint distribution of F = X and G = X + Y

X,Y,G,F are random variables

## Homework Equations

The vector [tex] W = (X, Y)^{t} [/tex] is a 2x1 multivariate Gaussian random vector with zero mean and covariance matrix equal to [tex]\Sigma[/tex], where [tex]\Sigma_{1,1}=1,\Sigma_{1,2} = \alpha, \Sigma_{2,1} = \alpha, \Sigma_{2,2} = 1.[/tex]

where [tex]|\alpha| < 1 [/tex]

## The Attempt at a Solution

1.[tex] cov(X,Y- \alpha X) = E(XY)- \alpha E(X^{2}) - E(X)E(Y) + \alpha E(X)E(X) = E(XY) - \alpha E(X^{2}) = 0? [/tex]

This would prove what I want but cannot get the last two terms to zero. I figured if I got [tex]E(X^{2})=0[/tex] I could use Cauchy inequality to prove the other term is zero but I can't get there.

2. Not sure how to start. Any reference that you think might be helpful would be greatly appreciated. Thanks.

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