ECE 534 Random Processes Instructor: R. Srikant
Midterm Exam II
Spring ’07 Time: 7:00-8:30 pm
You are allowed one 8.500 ×1100 sheet (two pages) of handwritten notes.
No calculators or electronic devices are allowed.
Each problem is worth 20 points. There are five problems.
Please write your name here:
1. Let X={Xt, t ∈ R} be a WSS Gaussian process.
(a) Let Yt=Xt+Xt−2. Is Y={Yt, t ∈ R} a WSS process? Is it a stationary process? Clearly explain your
answer.
Xis W SS and Gaussian. This means Xis stationary and Gaussian. Now obviously E(Yt) = 2µXand
E(Yt1Yt2) = E((Xt1+Xt1−2)(Xt2+Xt2−2)) = 2RX(t1−t2)+RX(t1−t2+2)+RX(t1−t2−2) = RY(t1−t2).
Since RY(t1, t2) depends only on t1−t2,Yis WSS too. Since Xis Gaussian and Yis a linear function of
X,Yis also Gaussian. Thus, Yis also stationary.
(b) What is the necessary and sufficient condition on RXfor Xto be Markov process?
Given it is Gaussian and stationary, it will be Markov if and only if RX(τ) = Aexp(−α|τ|) + µ2
Xfor some
constants A > 0 and α≥0.
(c) If Xis Markov, is Yalso Markov ? Clearly explain your answer.
Let RX(τ) be given by the expression in part (b). Now, substituting this in the formula for RY(τ) from part
(a), we can immediately see that RYcannot be in the required form so Yis not Markov.
2. A two-dimensional spatial Poisson process is used to model the distribution of points on a two-dimensional
plane. For example, it could be used to model the distribution of wireless communication devices in
Champaign-Urbana. Let Nbe a two-dimensional Poisson process of intensity λ.Nis defined as follows:
(i) For any set B∈ R2, the number of points in B, denoted by NB, is a Poisson random variable with mean
equal to λ∗Area(B).
(ii) If S1, S2, . . . , Snare disjoint sets in R2, then NS1, NS2, . . . , NSnare mutually independent.
Answer the following questions:
(a) What is the probability that there are no points in a circle of radius rcentered at the origin?
Since the area of the circle is r2π, the probability of the event is equal to e−λr2π.
(b) Let Rbe the distance from the origin of the closest point to the origin. Find the cdf and pdf of R.
F(r) = P(R≤r) = 1 −P(R > r) = 1 −e−λr2π,so f(r) = 2λπre−λr2π.
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