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Exam 2 for ECE 534 Spring 2008, Exams of Electrical and Electronics Engineering

University of Illinois - Urbana-ChampaignElectrical and Electronics Engineering

The spring 2008 exam 2 for the electrical and computer engineering (ece) 534 course. The exam covers various topics in stochastic processes, including markov processes, poisson processes, brownian motion, and gaussian random processes. Students are required to determine properties of processes, find probabilities, and justify statements.

Typology: Exams

Pre 2010

Uploaded on 03/10/2009

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ECE 534 Spring 2008
April 16, 2008
Exam 2
•You have 1.5 hours to complete this exam.
•Don’t forget to put your name on the answer booklet.
•You are allowed 2 sheets of notes (8.5”×11’, both sides).
•Calculators laptop computers, PDA’s, etc. are not permitted.
•Maximum possible score is 50.
•Neatness counts, especially for partial credit towards incorrect solutions.
1. (7 pts) Consider the WSS process {Xt:t∈R}with autocorrelation function:
RX(τ) = e−τ2
(a) (1 pt) Determine whether or not {Xt}is m.s. continuous.
(b) (3 pts) Determine whether or not {Xt}m.s. differentiable. If you conclude that it is m.s.
differentiable, find the mean and autocorrelation function of the derivative process.
(c) (2 pts) Find the autocovariance function CX(τ).
(d) (1 pts) Determine whether or not {Xt}is mean ergodic in the m.s. sense.
2. (10 pts) Let {Nt:t≥0}be a Poisson process with parameter λ= 1.
(a) (5 pts) Find P(N2≥2|N3= 3).
(b) (3 pts) Find P(N3= 3|N2≥2).
(c) (2 pts) Let Yt=√Nt. Determine whether or not {Yt}is a Markov process?
3. (12 pts) Let {Wt:t≥0}be a Brownian motion with parameter σ2= 1.
(a) (2 pts) Find E[W4|W3
2= 27]
(b) (4 pts) Find E[W3
2|W1= 2].
(c) (6 pts) Now suppose we define the random variable Yvia the m.s. integral
Y=Z1
0
t Wtdt
Find P{Y > 1}.
c
V. V. Veeravalli, 2008 1
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ECE 534 Spring 2008

April 16, 2008

Exam 2

  • You have 1.5 hours to complete this exam.
  • Don’t forget to put your name on the answer booklet.
  • You are allowed 2 sheets of notes (8.5”×11’, both sides).
  • Calculators laptop computers, PDA’s, etc. are not permitted.
  • Maximum possible score is 50.
  • Neatness counts, especially for partial credit towards incorrect solutions.
  1. (7 pts) Consider the WSS process {Xt : t ∈ R} with autocorrelation function: RX (τ ) = e−τ^2 (a) (1 pt) Determine whether or not {Xt} is m.s. continuous. (b) (3 pts) Determine whether or not {Xt} m.s. differentiable. If you conclude that it is m.s. differentiable, find the mean and autocorrelation function of the derivative process. (c) (2 pts) Find the autocovariance function CX (τ ). (d) (1 pts) Determine whether or not {Xt} is mean ergodic in the m.s. sense.
  2. (10 pts) Let {Nt : t ≥ 0 } be a Poisson process with parameter λ = 1. (a) (5 pts) Find P(N 2 ≥ 2 |N 3 = 3). (b) (3 pts) Find P(N 3 = 3|N 2 ≥ 2). (c) (2 pts) Let Yt = √Nt. Determine whether or not {Yt} is a Markov process?
  3. (12 pts) Let {Wt : t ≥ 0 } be a Brownian motion with parameter σ^2 = 1. (a) (2 pts) Find E[W 4 |W 23 = 27] (b) (4 pts) Find E[W 23 |W 1 = 2]. (c) (6 pts) Now suppose we define the random variable Y via the m.s. integral

Y =

0

t Wtdt

Find P{Y > 1 }.

©cV. V. Veeravalli, 2008 1

  1. (11 pts) Let {Xk : k = 1, 2 ,.. .} be a discrete-time random process with components that are i.i.d. Gaussian random variables with mean 1 and variance 1, i.e., Xk ∼ N (1, 1). Now define a new discrete-time random process {Yk : k = 1, 2 ,.. .} by

Yk =

∏^ k i=

Xi, k = 1, 2 ,...

(a) (3 pts) Determine whether or not {Yk} is Markov. (b) (3 pts) Determine whether or not {Yk} is a martingale. Hint: It is enough to check whether or not E[Yk+1|Y 1 · · · Yk] = Yk for all k ≥ 1. (c) (5 pts) Determine whether or not {Yk} is an independent increment process.

  1. (10 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) The function R(τ ) = (1 − |τ |)1 (^1) {|τ |≤ 3 } (note the range of the indicator function) is a valid autocorrelation function for a WSS process. (b) If {Xt} and {Yt} are jointly WSS processes, then the process {Zt} defined by Zt = Xt + Yt is also WSS. (c) Consider a WSS process {Xt} that is mean ergodic in the m.s. sense. If RX (τ ) converges as τ → ∞, then it must be the case that limτ →∞ RX (τ ) = 0. (d) If {Xt : t ∈ R} is a stationary Gauss-Markov process, then the process {Yt} defined by Yt = Xt + X 2 t is also a stationary Gauss-Markov process. (e) Suppose {Xt} is zero-mean, WSS and m.s. differentiable, and let derivative process be denoted by {X t′}. Then for fixed t, the random variables Xt and X t′ are uncorrelated.

©cV. V. Veeravalli, 2008 2