ECE 534 August 27, 2009
Fall 2009
Homework Assignment 1
Due Date: Thursday, September 10 (in class) Announcement: There will be a probability review quiz on Tuesday, September 15 from 8:009:30 PM (the room will be announced later). The quiz will be close
ECE 534 RANDOM PROCESSES
SOLUTIONS TO PROBLEM SET 1
FALL 2011
1 Some events based on rolls of three dice
(a) = cfw_x1 x2 x2 : 1 xi 6 for 1 i 3, or = cfw_111, 112, 113, 114, 115, 116, 121, 122, . . . , 664, 665, 666,
or = cfw_1, 2, 3, 4, 5, 63 .

5
(b) A
University of Illinois
Spring 2014
ECE 534: Homework 1
Issued: January 28th, 2014; Due: February 11th, 2014
Homework is due at the beginning of the lecture.
1. Probability Spaces. Suppose that = [0, 1] is the unit interval, and that F is the set
of all su
ECE 534 September 15, 2009
Fall 2009
Homework Assignment 2
Due Date: Thursday, September 24 (in class) Reading: Chapter 2 of text and the solutions to the even numbered problems of Chapter 2 given at the end of the book. Also, if you are not comfortable w
ECE 534 RANDOM PROCESSES
SOLUTIONS TO PROBLEM SET 1
FALL 2013
1 Some events based on a roll of three dice
(a) = cfw_x1 x2 x2 : 1 xi 6 for 1 i 3, or = cfw_111, 112, 113, 114, 115, 116, 121, 122, . . . , 664, 665, 666,
or = cfw_1, 2, 3, 4, 5, 63 .

5
(b) A
ECE 534 RANDOM PROCESSES
FALL 2013
SOLUTIONS TO PROBLEM SET 2
1 Convergence of alternating series
(a) For n 0, let In denote the interval with endpoints Sn and Sn+1 . It suces to show that
I0 I1 I2 . If n is even, then In = [Sn+1 , Sn+1 + An+1 ] [Sn+1 , S
ECE 534 RANDOM PROCESSES
FALL 2013
SOLUTIONS TO PROBLEM SET 3
1 Comparison of MMSE estimators for an example
(a) The minimum MSE estimator of X of the form g (Y ) is given by g (u) = exp(u), because this
estimator has MSE equal to zero. That is, E [X Y ]
University of Illinois
Fall 2012
ECE 534: Quiz
Wednesday September 19, 2012
6:00 p.m. 7:30 p.m.
116 Roger Adams Laboratory
Name:
University NetID:
You have one hour and 30 minutes for this quiz. The quiz is closed book and closed note.
Calculators, lapt
University of Illinois
Fall 2012
ECE 534: Quiz
Wednesday September 19, 2012
6:00 p.m. 7:30 p.m.
116 Roger Adams Laboratory
1. [10 points] Alice, Bob and Eve have two standard decks of playing cards. Every day, they
play the following game. Alice takes the
University of Illinois
Spring 2014
ECE 534: Homework 2
Issued: February 16th, 2014; Due: February 27, 2014
Homework is due at the beginning of the lecture.
1. Find the characteristic function of a standard Gaussian random variable X. Then, nd
the mean and
l
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University of Illinois
Spring 2014
ECE 534: Midterm I
March 4th, 2014
1. Problem 1. [25] a) The Halting Problem is the canonical undecidable problem in
computation theory that was rst introduced by Alan Turing in his seminal 1936 paper.
The problem is to
University of Illinois
Fall 2011
ECE 534: Exam II
Monday November 14, 2011
7:00 p.m. 8:15 p.m.
103 Talbot
1. [15 points] Suppose Xt = V exp(U t) where U and V are independent, and each is uniformly
distributed over the interval [0, 1].
(a) Find the mean f
University of Illinois
Fall 2012
ECE 534: Exam II
Wednesday November 28, 2012
6:00 p.m. 7:30 p.m.
165 Everitt Laboratory
1. [25 points] Consider the following continuoustime Markov process Y = (Yt )t0 where each
Yt takes values in cfw_0, 1. The distribut
ECE 534: Random Processes
Spring 2013
Midterm II
R. Srikant
Apr 18 2013, 7:008:15pm
Each question is worth 25 points. You are allowed one 8.5 11 sheet (two pages) of handwritten notes. You have to provide a clear reasoning to receive credit for any quest
ECE 534 RANDOM PROCESSES
SOLUTIONS FOR PROBLEM SET 6
FALL 2012
1 (7.1) Calculus for a simple Gaussian random process
(a): In order to verify this directly, we have to use the denition. We rst calculate Xs Xt which
st
is equal to B + C (t + s). Now, we hav
ECE 534: Random Processes
Spring 2013
Solutions for H.W I
R. Srikant
1.1 Simple events
(a) = cfw_0, 18 , or = cfw_x1 x2 x3 x4 x5 x6 x7 x8 : xi cfw_0, 1 for each i. It is natural to let F be the set of all
A
subsets of . Finally, let P (A) = 256 , where
ECE 534: Random Processes
Spring 2013
Solutions for H.W II
R. Srikant
2.7 Convergence of random variables on (0,1]
(a) To nd the CDF of Xn it is helpful to rst graph Xn . See Figure 1.
Figure 1: Xn , Fxn , and Xn X2n
The CDF is given by
FXn (c)
0, if c 0
ECE 534: Random Processes
Spring 2013
Solutions for H.W III
R. Srikant
3.1 Rotation of a joint normal distribution yielding independence
(a) Cov(X ) = 1 and Cov(X )1 =
1
2
1
1
so
T
(b) det
2
1
5
2
1
1
5
2
1
v1
v2
1
1 5
2
=
1
1
exp
2
2
=
fX ( x)
1
1
exp
ECE 534: Random Processes
Spring 2013
Solutions for H.W IV
R. Srikant
4.15 Invariance of properties under transformations
(a) FALSE. For example, let P [X0 = i] = 1 for i cfw_1, 0, 1 and let the sample paths all have the
3
form . . . , 1, 1, 1, 0, 1, 1, 0
ECE 534: Random Processes
Spring 2013
Solutions for H.W V
R. Srikant
8.1 On ltering a WSS random process
(a) True. Since SY ( ) = H ( )2 SX ( ), it follows that SY ( ) SX ( ) for all . Integrating over all
frequencies yields the result, because the powe
ECE 534: Random Processes
Spring 2013
HW VII
R. Srikant
1. Consider a discretetime queueing system with Ber() arrivals and Ber() service with < .
Assume departures occur before arrivals in each time slot.
(a) Write down the queueing dynamics of this syst
ECE 534 RANDOM PROCESSES
SOLUTIONS TO PROBLEM SET 1
FALL 2012
1. Persistence and coin tossing
(a) If 0 denotes T and 1 denotes H, then consists of all nite binary strings of length at least two, which
end in 11 and otherwise consist of 0s and 10s in vario
ECE 534 RANDOM PROCESSES
SOLUTIONS FOR SPROBLEM SET 2
FALL 2012
1 Stochastic convergence of sample variance
(a) We want to prove that E[V n ] = 2 .
E[V n ] =
=
1
E[
n1
1
E[
n1
1
E[
=
n1
=
=
=
=
1
n1
n
(Xi X n )2 ]
i=1
n
(Xi + X n )2 ]
i=1
n
(Xi )2 + (X n
ECE 534 RANDOM PROCESSES
SOLUTIONS FOR PROBLEM SET 3
FALL 2012
1 (3.1) Rotation of a joint normal distribution yielding independence
(a) This part can be done just by writing the formula (3.8) for joint gaussian distribution and
multiply the matrices:
fX
ECE 534 RANDOM PROCESSES
SOLUTIONS FOR PROBLEM SET 4
FALL 2012
1 (4.3) A sinusoidal random process
For the mean function we write:
X (t) = E [A cos(2V t + )]
= E[A]E
cos(2V t + )f ()d
[0,2 ]
=2E
cos(2V t + )
[0,2 ]
d
2
=0
For the Autocorrelation function:
ECE 534 RANDOM PROCESSES
SOLUTIONS FOR PROBLEM SET 5
FALL 2012
1 (4.31) Mean hitting time for a discretetime, discretestate Markov process
(a) The transition probability matrix P can be written as:
.6 .4 0
P = .8 0 .2
0 .4 .6
(b) In order to get the in
University of Illinois
Fall 2011
ECE 534: Final Exam
Friday, December 16, 2011
7:00 p.m. 10:00 p.m.
103 Talbot Laboratory
Name:
University NetID:
You have three hours for this exam. You may consult both sides of three 8.5 11 sheets of
notes. Otherwise th
ECE 534 RANDOM PROCESSES
PROBLEM SET 1
FALL 2004
Due September 8
Please visit the course website soon: courses.ece.uiuc.edu/ece534/
1. Review of Basic Probability
Assigned reading: Chapter one, Getting started, of the course notes, including the problems
ECE 534 RANDOM PROCESSES
PROBLEM SET 5
FALL 2012
Due beginning of class, Thursday, November 29
5. Markov Processes and All That
Assigned Reading: Chapter 4 and Sections 13 of Chapter 6 of the course notes.
Problems to be handed in:
Problems 4.31 and 4.37
University of Illinois at UrbanaChampaign
ECE 434: Random Processes
Fall 2004
Probability Quiz
Monday, September 13, 2004
Name:
You have one hour for this quiz. The quiz is closed book and closed note.
Calculators, laptop computers, Palm Pilots, twowa
ECE 534 RANDOM PROCESSES
PROBLEM SET 1
FALL 2012
Due beginning of class, Thu Sep 13
Please visit the course website soon: http:/maxim.ece.illinois.edu/fall12/
1. Review of Basic Probability
Assigned reading: Chapter 1 of the ECE 534 course notes, includin
ECE 534 RANDOM PROCESSES
PROBLEM SET 2
FALL 2012
Due beginning of class, Thursday, October 11
2. Convergence of a Sequence of Random Variables
Assigned Reading: Chapter 2 of the course notes and supplemental material to be posted on the
course webpage.
Re