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.