ECE 534 Spring 2008
May 5, 2008
Final Exam
•You have 2.5 hours to complete this exam.
•Don’t forget to put your name on the answer booklet.
•You are allowed 3 sheets of notes (8.5”×11”, both sides).
•Calculators laptop computers, PDA’s, etc. are not permitted.
•Maximum possible score is 90.
•Neatness counts, especially for partial credit towards incorrect solutions.
•You may find the following Fourier transform pairs to be useful:
e−at11{t≥0}↔1
a+jω and e−a|t|↔2a
a2+ω2,for a > 0
1. (20 pts, equally weighted parts) True or False. Determine if the following statements are True or
False. You need to justify your answer clearly to get credit – provide a short proof if you say the
statement is True, and a counter-example if you say the statement is False. Just stating “True”
or “False” without any justification will get zero credit.
(a) Suppose the joint characteristic function of the random variables X1and X2satisfies
ΦX1,X2(u1, u2) = 1
1−j(u1+u2)−u1u2
Then E[X1] = Var(X1) = 1.
(b) If Xand Yare random variables with finite second moments, then
E[(X−E[X|Y])2]≤E[(X−ˆ
E[X|Y, Y 2])2]
(c) If E[(X−E[X|Y])2] = Var(X), then it is necessarily the case that Xand Yare independent.
(d) If Xis a non-negative random variable with mean 1, then E[log X]≤0.
(e) If {Xt}and {Yt}are jointly WSS processes, then the process {Zt}defined by Zt=XtYtis
also WSS.
(f) R(τ) = (1 − |τ|)11{|τ|≤1}is a valid autocorrelation function for a WSS process.
(g) If the PSD of a WSS process {Xt}is given by SX(ω) = (1 − |ω|)11{|ω|≤1}, then E[X2
t] = 1.
(h) If U1, U2,..., is a sequence i.i.d. Unif[0,1] random variables and Xn= max{U1,...,Un},
n≥1, then Xnconverges in probability as n→ ∞.
(i) Suppose U∼Unif[0,1] and Xn= cos(2nπU ), n≥1, then Xnconverges in probability as
n→ ∞.
c
V. V. Veeravalli, 2008 1
