4. (3 points) Suppose that X, Y, and Z are multivariate Gaussian random variables with zero means and

the covariance matrix

Σ =

(a) Given that Y = 1 and Z

this MMSE estimate.

0² OXY σχε

OYZ

σYX σ}

62x σαν σε

-1, find an MMSE estimate of X. Compute the conditional MSE of

3 1 -1

-1 1 2

(3)

(b) Given that Y + 2Z = 2, find an MMSE estimate of X. Compute the conditional MSE of this

MMSE estimate.