Bivariate Normal Distribution with Covariance Matrix and Linear Transformation

[autobot.d]

Homework Statement

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

$$\Gamma$$
which is a 2x2 matrix whose entries are
$$\Gamma_{1,1} = 1$$
$$\Gamma_{1,2} = \alpha$$
$$\Gamma_{2,1} = \alpha$$
$$\Gamma_{2,2} = 1$$

with $$| \alpha | < 1$$

a) Find the joint distribution of $$W_1 = X$$ and
$$W_2 = X+Y$$

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.

 

[Ray Vickson]

Homework Statement

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

$$\Gamma$$
which is a 2x2 matrix whose entries are
$$\Gamma_{1,1} = 1$$
$$\Gamma_{1,2} = \alpha$$
$$\Gamma_{2,1} = \alpha$$
$$\Gamma_{2,2} = 1$$

with $$| \alpha | < 1$$

a) Find the joint distribution of $$W_1 = X$$ and
$$W_2 = X+Y$$

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.

Just use the bivariate moment-generating function
$$M(w_1,w_2) = E \exp(w_1 W_1 + w_2 W_2) = \int \int e^{w_1 v_1 + w_2 v_2} f_{W_1 W_2}(v_1,v_2) \: dv_1 \: dv_2.$$ See, eg.,
http://www.public.iastate.edu/~maitra/stat501/lectures/MultivariateNormalDistribution-II.pdf