ECE 534 RANDOM PROCESSES FALL 2008
PROBLEM SET 3 Due Wednesday, October 8
3. Random Vectors and Minimum Mean Squared Error Estimation
Assigned Reading: Chapter 3 and the section on matrices in the Appendix, in the course notes.
Problems to be handed in:
1 Comparison of MMSE estimators for an example
Let X=1
1+U, where Uis uniformly distributed over the interval [0,1].
(a) Find E[X|U] and calculate the MSE, E[(X−E[X|U])2].
(b) Find b
E[X|U] and calculate the MSE, E[(X−b
E[X|U])2].
2 Some identities for estimators, version 2
Let X, Y, and Zbe random variables with finite second moments and suppose Xis to be estimated.
For each of the following, if true, give a brief explanation. If false, give a counter example.
(a) E[(X−E[X|Y])2]≤E[(X−b
E[X|Y, Y 2])2].
(b) E[(X−E[X|Y])2] = E[(X−b
E[X|Y, Y 2]2] if Xand Yare jointly Gaussian.
(c) E[ (X−E[E[X|Z]|Y])2]≤E[(X−E[X|Y])2].
(d) If E[(X−E[X|Y])2] = Var(X),then Xand Yare independent.
3 Some simple examples
Give an example of each of the following, and in each case, explain your reasoning.
(a) Two random variables Xand Ysuch that b
E[X|Y] = E[X|Y],and such that E[X|Y|is not
simply constant, and Xand Yare not jointly Gaussian.
(b) A pair of random variables Xand Yon some probability space such that Xis Gaussian, Yis
Gaussian, but Xand Yare not jointly Gaussian.
(c) Three random variables X, Y, and Z, which are pairwise independent, but all three together are
not independent.
4 Conditional Gaussian comparison
Suppose that Xand Yare jointly Gaussian, mean zero, with Var(X) = Var(Y) = 10 and
Cov(X, Y ) = 8.Express the following probabilities in terms of the Qfunction.
(a) pa4
=P{X≥2}.
(b) pb4
=P[X≥2|Y= 3].
(c) pc4
=P[X≥2|Y≥3].(Note: pccan be expressed as an integral. You need not carry out the
integration.)
(d) Indicate how pa, pb,and pcare ordered, from smallest to largest.
5 Representation of three random variables with equal cross covariances
Let Kbe a matrix of the form
K=
1a a
a1a
a a 1
,
1
