EC 381
Fall, 2013
Homework 7
Due FRIDAY November 8, 2013, by 6:00 PM
1. Let U and V be jointly Gaussian, independent random variables with mean 0 and variance 1. Dene
X = U + V, Y = U 2V .
(a) Are X, Y jointly Gaussian?
(b) What are the expected value of
EC 381
Fall, 2013
Homework 4
Solutions
1. Assume that Y is an exponential random variable, with V ar[Y ] = 36.
(a) Compute the pdf fY (y) of Y
(b) Compute E[Y 2 ]
(c) Compute P [Y < 6]
Solution:
(a) For an exponential random variable with parameter , the
EC 381 Probability Theory in ECE
Exam 1 Review
Prof. Bobak Nazer
Boston University
October 6, 2015
Foundations of Probability
Sets:
A set is a collection of elements.
The universal set S is the set of all elements (for the specific
context).
A subset A
Probability and Stochastic Processes
A Friendly Introduction for Electrical and Computer Engineers
Third Edition
MARKOV CHAINS SUPPLEMENT (MCS)
Roy D. Yates
David J. Goodman
February 3, 2014
This supplement uses a page size matched to the screen of an iPa
EC381 Probability Theory in ECE
Prof. Vivek Goyal
Boston University, Fall 2016
September 30, 2016
Homework 3 Solutions
Problem 3.1 The year is 1987, and you have been hired to work as a game programmer for the
Macintosh SE. The Macintosh SE supports eight
EC381 Probability Theory in ECE
Prof. Vivek Goyal
Boston University, Fall 2016
September 16, 2016
Homework 1 Solutions
Problem 1.1 Let A, B, C be three events in a sample space S. Each of the statements below
describes an event built from events A, B, and
EC381 Probability Theory in ECE
Prof. Vivek Goyal
Boston University, Fall 2016
September 17, 2016
Homework 2
Due: Friday, September 23, 5:00pm (in the box outside PHO 435)
Problem 2.1 You would like to evaluate the probability of success for testing a bat
EC 381
Fall, 2013
Homework 1
Due Date: Wed., Sep. 18, 2013
1. Below is a gure depicting the universal set S, and two subsets A, B. For each of the questions that follow,
shade the set in the gure that would result from the specied operations.
S
A
B
S
S
A
EC381 Probability Theory
c
2016
David Casta
n
on
Boston University, Spring 2016
Feb. 22
Homework 4
Due: 5pm, Friday Feb. 26 (in the box outside PHO434)
Reading: Yates & Goodman Chapter 4, and Fall 2014 Slides: Continuous Random Variables.
Problem 4.1 Assu
EC381 Probability Theory
c
2016
David Casta
n
on
Boston University, Spring 2016
March 1
Homework 5
Due: 5pm, Friday March 4 (in the box outside PHO434)
Reading: Yates & Goodman Chapters 4, 5, and Fall 2014 Slides: Continuous Random
Variables and Pairs of
EC381 Probability Theory
c
2016
David Casta
n
on
Boston University, Spring 2016
Feb. 5
Homework 3
Due: 5pm, Friday Feb. 12th (in the box outside PHO434)
Reading: Yates & Goodman Chapter 3, and Fall 2014 Slides: Discrete Random Variables.
Problem 3.1 Let X
EC381 Probability Theory
c
2016
David Casta
n
on
Boston University, Spring 2016
January 21
Homework 1
Due: 5pm, Friday January 29th (in the box outside PHO434)
Reading: Yates & Goodman Chapter 1.
Fall 2014 Slides: Foundations of Probability.
Problem 1.1 T
EC381 Probability Theory
c
2016
David Casta
n
on
Boston University, Spring 2016
March 14
Homework 6
Due: 5pm, Friday March 18 (in the box outside PHO434)
Reading: Yates & Goodman Chapters 5, 7 and Fall 2014 Slides: Pairs of Random Variables
Problem 6.1 Co
EC381 Probability Theory
c
2016
David Casta
n
on
Boston University, Spring 2016
January 29
Homework 2
Due: 5pm, Friday Feb. 5th (in the box outside PHO434)
Reading: Yates & Goodman Chapters 2. 3, and Fall 2014 Slides: Foundations of Probability
and Discre
EC 381 Probability Theory in ECE
Exam 2 Review
Prof. Bobak Nazer
Boston University
November 12, 2015
Continuous Random Variables
Cumulative Distribution Function (CDF):
This is the probability that a random variable X is less than or equal
to a value x:
EC 381
Fall, 2012
Exam 2
Solutions
1. (22 pts) Indicate whether each of the following statements is true or false by clearly writing true or false.
Explain briey if you want to get partial credit for a wrong answer.
(a) Let X, Y be uncorrelated random var
EC 381
Fall, 2012
Exam 1
Thursday, Oct. 11, 2012
1. (48 pts) Indicate whether each of the following statements is either always true or sometimes false by
clearly writing true or false. Explain briey if you want partial credit. Diagrams are welcome.
Let A
EC 381
Fall, 2012
Homework 2
Due Date: Wednesday, September 25, 2013
1. A memory module consists of 2 chips. The memory module is designed with redundancy so that it works
even if one of its chips is defective. Each chip consists of n transistors, and the
EC 381
Fall, 2013
Homework 3
Due Date: Wed, Oct. 2, 2013
1. A random variable n has pmf
PN (n) =
c(1/2)n
0
n = 0, 1, 2
otherwise
(a) What is the value of the constant c?
(b) Compute the cumulative distribution function (cdf).
(c) Compute E[N ].
(d) Comput
EC 381
Fall, 2013
Homework 4
Due Wednesday, Oct 9, 2013.
1. Assume that Y is an exponential random variable, with V ar[Y ] = 36.
(a) Compute the pdf fY (y) of Y
(b) Compute E[Y 2 ]
(c) Compute P [Y < 6]
2. Let X be a uniform random variable distributed ov
EC 381
Fall, 2013
Homework 5
Due Wednesday, October 23, 2013
1. Consider a factory that produces circuits that are good with reliability p = 0.9, independently from circuit
to circuit. Assume that you select 3 circuits to test. Dene the random variable X
EC 381
Fall, 2013
Homework 6
Due Friday November 1, 2013
1. The random variable X is exponentially distributed with parameter x = 1; in other words, continuous
random variable X is dened by the PDF
fX (x) = ex ,
x0
Given that X = x, the random variable Y
EC 381
Fall, 2013
Homework 8
Due: Solutions
1. Consider a radar system summarized as follows: Under hypothesis H0 , the return signal Y = W , where
W N (0, 1). Under hypothesis H1 , the return signal is Y = 2 + W . The prior probabilities of H0 , H1 are
a
EC 381
Fall, 2013
Homework 9
Due: December 11, 2013
1. Consider the following problem. You have a fair coin, with probability 0.5 of Heads or Tails. You are going
to ip the coin until you get 100 heads. The goal is to compute the probability that, when yo
EC 381
Fall, 2013
On-line HW 1 Answers
1. Let A and B be arbitrary events with 0 < P (A), P (B) < 1. Indicate whether each of the following statements
is true always as written or false by clearly writing true or false. Draw a diagram for each question
il
EC 381
Fall, 2013
Homework 1
Solutions
1. Below is a gure depicting the universal set S, and two subsets A, B. For each of the questions that follow,
shade the set in the gure that would result from the specied operations.
S
A
B
S
S
A
AB
S
S
A
(Ac B) (A B
EC 381
Fall, 2012
Homework 2
Take Home Solutions
1. A memory module consists of 2 chips. The memory module is designed with redundancy so that it works
even if one of its chips is defective. Each chip consists of n transistors, and the chip is built with
EC 381
Fall, 2013
Homework 3
Solutions
1. A random variable n has pmf
PN (n) =
c(1/2)n
0
n = 0, 1, 2
otherwise
(a) What is the value of the constant c?
Solution: Note the following
2
(1/2)n =
n=0
Hence, in order for
2
n=0
7
2
4
PN (n) = 1, c = 4/7.
(b) Co
EC 381
Fall, 2013
Homework 7
Solutions
1. Let U and V be jointly Gaussian, independent random variables with mean 0 and variance 1. Dene
X = U + V, Y = U 2V .
(a) Are X, Y jointly Gaussian?
Solution: They are, because sums of Gaussians are Gaussians, and
EC 381
Fall, 2013
Exam 2
Wednesday, November 20, 2013
1.
2.
3. (10 pts)
4. (12 pts) On the basis of a sensor output X , it is to be decided which hypothesis is true: H0 or H1 . If H0 is
true, then X is a discrete random variable taking values in cfw_1, 2,