3. (3 points) Suppose that [X Y Z] is a multivariate Gaussian random vector:

1 0.5 1

B-GED

~N

0.5 1 0

0 2

X

(1)

(a) Obtain the joint distribution of X and Z.

(b) Find an MMSE estimate of X given Z = 1. What is the conditional mean squared error (MSE)

of this MMSE estimate?

(e) Obtain the joint distribution of X-Y and Z.

Hint: note that

[***]-6 #1 10

Z

0 0

(d) Find an MMSE estimate of X - Y given Z= 1. What is the conditional MSE of this estimate?

(2)