ECE 534 Spring 2008
February 20, 2008
Homework Assignment 3
Due Date: Monday, March 3 (in class)
Reading: Chapter 3 of text. Also read the solutions to the even numbered problems of Chapter 3
given at the end of the book.
1. Problem 3.1.
2. Problem 3.3. Hint for (b): Can E[X|Y] be negative.
3. Problem 3.5.
4. Problem 3.7. Note: Regarding the uniqueness in part (b), the answer is non-unique. You may
want to think of an example to show that the answer is non-unique, but you don’t have to do it
as part of the homework.
5. Using the joint characteristic function show that if X1,X2,X3,X4are jointly Gaussian (correlated)
random variables with zero mean, then
E[X1X2X3X4] = E[X1X2]E[X3X4] + E[X1X3]E[X2X4] + E[X1X4]E[X2X3]
6. Problem 3.9.
7. Problem 3.15.
8. (Problem 3.17 rewritten with better notation.) Assume that Xand Zare independent Gaussian
vectors with means µX,µZand covariance matrices KXand KZ. Let Y=X+Z. Then clearly
Xand Yare jointly Gaussian.
(a) Show that
E[X|Y] = µX+KX(KX+KZ)−1(Y−(µX+µZ))
=KZ(KX+KZ)−1µX+KX(KX+KZ)−1(Y−µZ)
(b) Show that the conditional covariance matrix of X, given Yis given by
Cov(e) = KX−KX(KX+KZ)−1KX=KX(KX+KZ)−1KZ= (K−1
X+K−1
Z)−1
where e=X−E[X|Y]
c
V. V. Veeravalli, 2008 1