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 <
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 statemen
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 i
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,
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) F
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 t
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 t
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:
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 201
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 programme
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
EC381
Probability
(and a little Statistics)
Course Information
Instructor
Prof. Prakash Ishwar ([email protected], Pho 440)
Office Hours: Wed 9:30-11am + 5-6:30pm + email for extra help
Overview
David Cast
EC381 Probability Theory
c
David
Casta
n
on
Boston University, Spring 2016
March 18
Online Quiz 6 Solutions
Reading: Yates & Goodman Chapter 5, 7
Fall 2014 Slides: Multiple Random Variables.
Question
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
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 Fal
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
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 20
9/28/2014
SCOPE OF THE COURSE
EC381
Probability
(and a little Statistics)
Probability description of
Random Events
Random Variables
Overview
Random Functions
David Castan
[email protected]
Prakash Ishwar [email protected]
EC381 Probability Theory
Boston University, Fall 2017
Homework 2 Solutions
Reading: Yates & Goodman Chapter 1,2.
Fall 2014 Slides: Foundations of Probability.
Problem 2.1 We would like to determine th
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
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 wri
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 head
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
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
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
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 dis
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
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
Henc
EC381 Probability Theory: II. Discrete Random Variables
II.1
1
Discrete Random Variables
A random variable is a mapping that assigns real numbers to outcomes in the sample space.
Random variables ar
EC381 Probability Theory
Boston University, Fall 2017
Homework 1 Solutions
Reading: Yates & Goodman Chapter 1.
Fall 2014 Slides: Foundations of Probability.
Problem 1.1 Let A, B, C be three events in
EC381 Probability Theory: I. Foundations of Probability
I.1
1
Set Theory
Probability theory is built upon set theory. This is a very brief primer.
A set is a collection of elements.
The empty set or
EC381 Concept List 1: Foundations of Probability
Set Theory
- Set, Universal Set, Subset
- Set Operations: Union, Intersection, Complement
- Mutually Exclusive (or Disjoint) Sets and Collectively Exh