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## Homework Statement

Z is a 2x1 multivariate gaussian random vector, where [tex] Z = (X Y)^t [/tex], X,Y are real numbers, with mean zero and covariance matrix

[tex] \Gamma[/tex]

which is a 2x2 matrix whose entries are

[tex] \Gamma_{1,1} = 1[/tex]

[tex] \Gamma_{1,2} = \alpha [/tex]

[tex] \Gamma_{2,1} = \alpha [/tex]

[tex] \Gamma_{2,2} = 1 [/tex]

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

a) Find the joint distribution of [tex] W_1 = X [/tex] and

[tex] W_2 = X+Y[/tex]

b) Find the conditional pdf of X+Y given X.

## The Attempt at a Solution

I think I want to do a linear transformation to get a) but not sure how to attack the problem. Any help/references would be greatly appreciated. This is easy for independent gaussian variables but this is not the case here.

Thanks.