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
1
HW 1 Solution
Problem 1.1 cfw_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 subsets of . Finally, let P (A) =
A , 256
where A denotes the cardinality of a set A.
ECE 534 RANDOM PROCESSES
PROBLEM SET 2
FALL 2004
Due September 22
Sequences of Random Variables
Assigned Reading: Chapter 2 and Sections 8.18.3 of the course notes. Additional material on
limits for deterministic sequences can be found in Kenneth Ross, E
ECE 534 RANDOM PROCESSES
SOLUTIONS TO PROBLEM SET 5
FALL 2011
1 Estimation of the parameter of an exponential in additive exponential noise
(a) By assumption, Z has the exponential distribution with parameter , and given Z = z, the
conditional distributio
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
University of Illinois
ECE 534: Homework 6
Issued: April 7th, 2012; Due: April 17th, 2012
Homework is due at the beginning of the lecture.
1. Problems 4.5, 4.7, 4.13, 4.17, 4.19, 4.21, 4.24, 4.31 from the textbook.
Spring 2012
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 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
ECE 534: Random Process
Fall 2014
Problem Set 4 Solution
R. Srikant
Due: Oct. 23rd
Question 1 (Problem 4.1 from Hajeks Note)
Solution:
(a) There are four sample functions corresponding to the four possible values of (A, B) as shown in
the gure below.
(b)
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
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
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
l
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M Ham MW mag, {a m, a (amok. o alh.
1)[.w:n 1w1; 1) :
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H
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Li. l, Lat
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
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
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
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
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 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: