UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 2
Fall 2015
Issued: Thursday, September 03, 2015 Due: 9am, Thursday, September 10, 2015
Problem 1. Consider a binary tree with n lev
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 6
Fall 2015
Issued: Thursday, October 8, 2015
Due: 9:00am, Thursday, October 15, 2015
Problem 1. A discrete-time Markov chain with s
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 8
Fall 2015
Issued: Thursday, October 22, 2015
Due: 9:00am, Thursday, October 29, 2015
Problem 1. Consider the continuous time Marko
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 9
Fall 2015
Issued: Thursday, November 5, 2015
Self-graded Scores Due: 5:00pm Monday, November 9, 2015
Submit your self-graded scores via
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 9
Fall 2015
Issued: Thursday, October 29, 2015
Due: 9:00am, Thursday, November 5, 2015
Problem 1. Given X = i, Y Exp(i ), for i = 0,
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 2
Fall 2015
Date: Wednesday, September 9, 2015
Problem 1. You have two envelopes, and you know that each contains a positive
integer
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 1
GSI : Kangwook Lee
Date: Wednesday, September 2, 2015
Problem 1. Consider the following chess boards. Note that a rook can move eit
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 2
Fall 2015
Date: Wednesday, September 9, 2015
Problem 1. Let n be the number in the larger envelope and m be the smaller one.
Dene t
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 1
GSI : Kangwook Lee
Date: Wednesday, September 2, 2015
Problem 1. Consider the following chess boards. Note that a rook can move eit
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 3
GSI: Kangwook Lee
Date: Wednesday, September 16, 2015
Problem 1. Let X Unif[0, 1]. If Y = 2X, what is the PDF of Y ?
Problem 2. Let
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 8
Fall 2015
Issued: Thursday, October 29, 2015
Self-graded Scores Due: 5:00pm Monday, November 2, 2015
Submit your self-graded scores via
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 7
Fall 2015
Issued: Thursday, October 15, 2015
Due: 9:00am, Thursday, October 22, 2015
Problem 1. Consider a Poisson process cfw_Nt
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 1
Fall 2015
Issued: Thursday, September 3, 2015
Self-graded Scores Due: 5pm, Monday, September 7, 2015
Submit your self-graded scores via
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 3
Fall 2015
Issued: Thursday, September 10, 2015 Due: 9am, Thursday, September 17, 2015
Problem 1. This problem will explore an impo
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 3
Fall 2015
Issued: Thursday, September 17, 2014
Self-graded Scores Due: 5:00pm Monday, September 21, 2015
Submit your self-graded scores
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 4
Fall 2015
Issued: Friday, October 2, 2014
Self-graded Scores Due: 5:00pm Monday, October 5, 2015
Submit your self-graded scores via the
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 4
Fall 2015
Issued: Thursday, September 24, 2015
Due: 9:00am Thursday, October 1, 2015
Problem 1. Midterm 01.
Problem 2. Suppose we
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 5
Fall 2015
Issued: Thursday, October 8, 2015
Self-graded Scores Due: 5:00pm Monday, October 12, 2015
Submit your self-graded scores via
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Problem Set 5
Fall 2015
Issued: Thursday, October 1, 2015
Due: 9:00am Thursday, October 8, 2015
Problem 1. Let X1 , X2 , . . . , Xn be n i.i.d.
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 7
Fall 2015
Issued: Thursday, October 22, 2015
Self-graded Scores Due: 5:00pm Monday, October 26, 2015
Submit your self-graded scores via
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution 6
Fall 2015
Issued: Thursday, October 15, 2015
Self-graded Scores Due: 5:00pm Monday, October 19, 2015
Submit your self-graded scores via
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 3
GSI: Kangwook Lee
Date: Wednesday, September 16, 2015
Problem 1. Let X Unif[0, 1]. If Y = 2X, what is the PDF of Y ?
Solution 1. We
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Processes
Discussion 5
Fall 2015
Date: Wednesday, September 30, 2015
1
Some Quick Notes
(1) For a random variable X with transform MX (s), the transform o
EECS 126 Probability and Random Processes
Kannan Ramchandran
University of California, Berkeley: Fall 2014
December 16, 2014
Final Exam
Last name
First name
SID
Rules.
DO NOT open the exam until instructed to do so.
Note that the test has 110 points. Th
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Final Solution
Fall 2014
Problem 1.
(a) First we nd the conditional pdf:
f (y1 . . . , yn |x) =
1 1
e x
xn
i
yi
.
Thus, XM AP = 1 if
pe
i
yi
1
> (
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution of Sample Midterm 2
Fall 2014
For the sample midterm, we write short solutions. You are supposed to
show your work in the actual exam.
Pr
EECS 126 Probability and Random Processes
Kannan Ramchandran
University of California, Berkeley: Fall 2014
November 7, 2014
(Practice Version) Midterm Exam 2
Last name
First name
SID
Name of student on your left:
Name of student on your right:
DO NOT ope
UC Berkeley
Department of Electrical Engineering and Computer Sciences
EE126: Probability and Random Process
Solution of Midterm 2
Fall 2014
Problem 1.
(a) We have
Pr(X > 4) = Pr(X 1 > 3)
1
= Pr(|X 1| > 3)
2
1
= 2/9 = 1/9.
2
(b) They have same distributio
EECS 126 Probability and Random Processes
Kannan Ramchandran
University of California, Berkeley: Fall 2014
November 13, 2014
Midterm Exam 2
Last name
First name
SID
Name of student on your left:
Name of student on your right:
DO NOT open the exam until i
Recitation 4
1. (a) Derive the expected value rule for functions of random variables E[g(X)] =
!
x g(x)pX (x).
(b) Derive the property for the mean and variance of a linear function of a random variable
Y = aX + b.
E[Y ] = aE[X] + b,
var(Y ) = a2 var(X).