ECE 434 RANDOM PROCESSES SPRING 2003
PROBLEM SET 4 Due Wednesday, March 19
Random Processes, including Poisson, Wiener, Markov and Martingale Processes
Assigned Reading: Chapter 4 of the notes.
Reminder: Exam I will be on Monday, March 10, 7:00 p.m. - 8:15 p.m. in Room 269 EL.
Problems to be handed in:
1. Let Xt=Acos(W t +Y) where A, W and Yare independent random variables, Ahas mean 2
and variance 4, Yis uniform on [−π,π] and Wis uniform on [0,5]. Find the mean function µX(t)
and autocorrelation function RX(s, t).Is Xwide sense stationary?
2. Let Yand Zbe independent random processes with
RY(s, t) = 2 exp(−|s−t|) cos(2πf(s−t)) and RZ(s, t) = 9+exp(−3|s−t|4).Find the autocorrelation
function RX(s, t) where Xt=YtZt.
3. Suppose that X1and X2are random variables such that EX1=EX2=E X1X2= 0 and
V ar(X1) = V ar(X2) = σ2. Define Yt=X1sin(t) + X2cos(t). (a) Is the random process Y
necessarily wide-sense stationary? (b) Give an example of random variables X1and X2satisfying
the given conditions such that Yis not (strict sense) stationary. (c) Give an example of random
variables X1and X2satisfying the given conditions such that Yis stationary.
4. Define a random process Xby Xt=A+Bt+t2, where Aand Bare independent, N(0,1) random
variables. (a) Find ˆ
E[X5|X1], the linear minimum mean square error (LMMSE) estimator of X5
given X1, and compute the mean square error. (b) Find the MMSE (possibly nonlinear) estimator
of X5given X1, and compute the mean square error. (c) Find ˆ
E[X5|X0, X1] and compute the mean
square error. (Hint: Can do by inspection.)
5. Define the random process Xby Xt= 2A+Bt where Aand Bare independent random variables
with P[A= 1] = P[A=−1] = P[B= 1] = P[B=−1] = 0.5. (a) Sketch the possible sample
functions. (b) Find P[Xt≥0] for all t. (c) Find P[Xt≥0 for all t].
6. Identification of special properties of two discrete time processes
Determine which of the properties:
(i) Markov property
(ii) martingale property
(iii) independent increment property
are possessed by the following two random processes. Justify your answers.
(a) X= (Xk:k≥0) defined recursively by X0= 1 and Xk+1 = (1 + Xk)Ukfor k≥0, where
U0, U1,... are independent random variables, each uniformly distributed on the interval [0,1].
(b) Y= (Yk:k≥0) defined by Y0=V0,Y1=V0+V1, and Yk=Vk−2+Vk−1+Vkfor k≥2, where
Vk:k∈Zare independent Gaussian random variables with mean zero and variance one.
7. Identification of special properties of two continuous time processes
Answer as in the previous problem, for the following two random processes:
(a) Z= (Zt:t≥0), defined by Zt= exp(Wt−σ2t
2), where Wis a Brownian motion with parameter
σ2. (Hint: Observe that E[Zt] = 1 for all t.)
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