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
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 8:00-9:30 PM (th
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
ECE 534 RANDOM PROCESSES
PROBLEM SET 2
FALL 2004
Due September 22
Sequences of Random Variables
Assigned Reading: Chapter 2 and Sections 8.1-8.3 of the course notes. Additional material on
limits for
ECE 534 RANDOM PROCESSES
FALL 2011
SOLUTIONS TO PROBLEM SET 3
1 Comparison of MMSE estimators for an example
(a) The minimum MSE estimator of X of the form g (U ) is given by g (u) = u3 , because this
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
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 pos
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 p
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
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, . . . ,
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
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 t
1
Homework 2 Solution
Problem 1.13 Poisson and geometric random variables with conditioning - i e- i p(1 - p)j-1 = e [(1-p)] = e-p (a) P [Y < Z] = i=0 i=0 j=i+1 i! i! i-1 - j (b) P [Y < Z|Z = i] = P [
University of Illinois
Fall 2011
ECE 534: Quiz
Monday September 12, 2011
7:00 p.m. 8:00 p.m.
269 Everitt Laboratory
1. [10 points] Consider the following two stage experiment. A number X is determined
University of Illinois
ECE 534: Homework 7
Issued: April 17th, 2012; Due: April 26th, 2012
Homework is due at the beginning of the lecture.
1. Problems 5.1, 5.2, 5.4, 5.5, 5.6, 5.7, and 5.12 from the
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
di
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 b
University of Illinois
Spring 2012
ECE 534: Homework 1
Issued: January 21st, 2012; Due: January 31st, 2012
Homework is due at the beginning of the lecture.
1. Suppose that = [0, 1] is the unit interva
University of Illinois
Spring 2012
ECE 534: Homework 8
Issued: April 27th, 2012; Due: May 4th, 2012
Homework is due at the beginning of the Final Exam.
1. Assume that (Xt : t T ) is a a) Poisson proce
University of Illinois
Spring 2012
ECE 534: Homework 1
Issued: February 29th, 2012; Due: March 13th, 2012
Homework is due at the beginning of the lecture.
1. Give a counterexample to each of the follo
University of Illinois
Spring 2012
ECE 534: Homework 1
Issued: February 10th, 2012; Due: February 23rd, 2012
Homework is due at the beginning of the lecture.
1. Let X be a random variable uniformly di
University of Illinois
Spring 2012
ECE 534: Homework 5
Issued: March 19th, 2012; Due: March 29th, 2012
Homework is due at the beginning of the lecture.
1. The Fibonacci sequence of integers equals 1,
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 Al
l-
Lek N: be, a. him": Wm vm.-~n.,b1a., m. ru 1 .f
M Ham MW mag, {a m, a (amok. o alh.
1)[.w:n 1w1; 1) :
Mm ma) : Nwr W1)iwrmmwr)
I "13%;: I' PLW: 1-1. lmlrlwlrl) prL:G|WI:-1)
H
«BE1* Piwgrler)
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 sta
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(2
ECE 534 RANDOM PROCESSES
SOLUTIONS FOR PROBLEM SET 5
FALL 2012
1 (4.31) Mean hitting time for a discrete-time, discrete-state Markov process
(a) The transition probability matrix P can be written as:
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 calculat
ECE 534: Random Processes
Spring 2013
Midterm II
R. Srikant
Apr 18 2013, 7:00-8:15pm
Each question is worth 25 points. You are allowed one 8.5 11 sheet (two pages) of handwritten notes. You have to pr
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, . . .
12/4/2017
ECE 534: Random Processes
ECE 534: RANDOM PROCESSES, FALL 2017
Class time and place: 11:00-12:20 TTh, 3017 ECEB
Instructor: Prof. Olgica Milenkovic 311 CSL, milenkov at illinois dot edu
Offi
ECE 534: Random Processes
Spring 2017
Quiz Solution
1. Problem 1
(a) (A B C c ) (A B c C) (Ac B C).
(b) A B C c .
(c)
P(A B C c ) = P(A)P(B)P(C c ) = P(A)P(B)(1 P(C).
(d)
P(A B C) = P(A) + P(B) + P(C)
ECE 534: SPRING 2017
PREPARATION QUIZ
ISSUED: FEBRUARY 6TH.
This is a self-examination quiz, and it is designed to help you evaluate your background
in elementary probability and your understanding of
ECE 534 RANDOM PROCESSES
PROBLEM SET 6
FALL 2008
Due Wednesday, November 19
6. Basic Calculus of Random Processes
Assigned Reading: Chapter 7 and Sections 11.3-11.5 of the notes.
Reminder: Exam 2, cov
ECE 534 RANDOM PROCESSES
PROBLEM SET 3
FALL 2008
Due Wednesday, October 8
3. Random Vectors and Minimum Mean Squared Error Estimation
Assigned Reading: Chapter 3 and the section on matrices in the App