A random vector (X, Y, Z) has joint density given by

. f (x, y,z) = k exp [− 1/2 (2x2 − 2xy + y2 + z2 + 2x − 6y) .

 

1. Compute k. 

2. Compute the expectations P(X), P(Y ) and P(Z). 

3. Compute the density of the random vector (X, Z). 

4. Compute the correlation coeffificient between X and Z and between X and Y . 

5. Let W = X + Z; compute the probability density of W.