Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 10
Measurability and Random Variables
1
EE 325 Probability and Random Processes
Aug 12,2013
Measurable Spaces
Recall the denition of a probability space (, F , P ). A sligh
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 5
Lecture Notes 3
1
EE 325 Probability and Random Processes
July 28, 2014
Axiomatic Probability
We have learned some paradoxes associated with traditional probability theor
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 4
Lecture Notes 2
1
EE 325 Probability and Random Processes
July 24, 2014
Classical Probability
The word classical in the title is a slight misnomer. The intention here is
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 8
Lecture Notes 5
1
EE 325 Probability and Random Processes
August 7, 2014
Constructing Probability Spaces
We will use our axioms and set properties to paint a larger pictu
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 14
Lecture Notes 9
1
EE 325 Probability and Random Processes
August 28, 2014
Variance of a Random Variable
The expectation E[X] is also known as the first moment or mean of
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 25
Lecture Notes 20
1
EE 325 Probability and Random Processes
November 5, 2014
Markovs Inequality
Recall the Markovs inequality for the discrete random variables. An exact
Indian Institute of Technology Bombay
Dept of Electrical Engineering
Quiz I
20 marks
EE 325 Probability and Random Processes
Sep 1,2014
Question 1) Consider a random variable X with distribution Q(x), x E. Here E is some
discrete state-space and Q(x) [0,
Indian Institute of Technology Bombay
Dept of Electrical Engineering
Quiz III
15 marks
EE 325 Probability and Random Processes
Nov 8, 2014
Question 1) Let us define the differential entropy of a continuous valued random vector
as h(X) = f (x) log f (x)dx
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 20
Lecture Notes 13
1
EE 325 Probability and Random Processes
September 22, 2014
Markov Chains
Till now, we have been dealing mostly with random variables and random vector
Indian Institute of Technology Bombay
Dept of Electrical Engineering
Quiz III
Tutorial Solutions
EE 325 Probability and Random Processes
Nov 19, 2014
Question 1) (Rohatgi2001) Consider a bicyclist who leaves a point P (see Figure), choosing one of the roa
Notes for ECE 534
An Exploration of Random Processes for Engineers
Bruce Hajek
January 1, 2014
c 2014 by Bruce Hajek
All rights reserved. Permission is hereby given to freely print and circulate copies of these notes so long as the notes are left
intact a
EE325 Tutorial V
Notation:
N (t) is the white Gaussian noise process.
W (t) is the Weiner process.
(t) is the Poisson process.
1. Let Y (t) and Z(t) be two independent WSS processes. Let X(t) = Y (t)Z(t).
Find the auto-correlation function for X(t) in
EE325 Tutorial I
1. Let F1 , F2 , . . . be the -algebras. Verify whether the following following
sets are -algebras:
1. F = n Fn
2. F = n Fn
3. F = F1 F2 , where F1 F2 = cfw_A B : A F1 and B F2 .
2. Consider a function g : 1 2 and let F be a -algebra on 1
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 24
Conditional Density and Expectation
1
EE 325 Probability and Random Processes
Oct 31,2013
Detection Theory
The conditional probability and expectation that we learned ha
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 19
Markov Chains
1
EE 325 Probability and Random Processes
Oct 7,2013
Markov Chains
Till now, we have been dealing mostly with random variables and random vectors. In
betwe
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 9
Borel Sets
1
EE 325 Probability and Random Processes
Aug 5,2013
Uniform on the Square
Let us consider the real interval in R2 , which is an exciting space for many applic
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 3
Lecture Notes 1
1
EE 325 Probability and Random Processes
July 18, 2013
Introduction
Some basics of probability theory are typically taught in senior high-school, and man
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 6
Axiomatic Probability
1
EE 325 Probability and Random Processes
July 29,2013
Axiomatic Probability
We have learned some paradoxes associated with traditional probability
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 4
Class 2
1
EE 325 Probability and Random Processes
July 22, 2013
Classical Probability
The word classical in the title is a slight misnomer. The intention here is to diere
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 11
Independence of Events
1
EE 325 Probability and Random Processes
Aug 19,2013
Independent Events
We have learned about events being disjoint or mutually exclusive. Indepe
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 7
Probability Space Construction
1
EE 325 Probability and Random Processes
Aug 1, 2013
Constructing Probability Spaces
We will use our axioms and set properties to paint a
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 15
Independence of Random Variables
1
EE 325 Probability and Random Processes
Sep 2,2013
Functions of Two Random Variables
Consider two random variables X1 and X2 , not nec
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 13
Discrete Random Variables
1
EE 325 Probability and Random Processes
Aug 26,2013
Discrete Random Variables
We now learn an important special case of random-variables, the
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 17
Generating Functions
1
EE 325 Probability and Random Processes
Sep 19,2013
Sum of Two Random Variables
We now consider the sum of two independent random variables, say X
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 21
Continuous-valued Random Variables
1
EE 325 Probability and Random Processes
Oct 28,2013
Continuous Valued Random Variables
Once we thoroughly understand discrete random
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 22
Gaussian Vectors, Characteristic Functions
1
EE 325 Probability and Random Processes
Oct 28,2013
Covariance Matrices
We have encountered the matrix K while dealing with
Indian Institute of Technology Bombay
Department of Electrical Engineering
Handout 23
Conditional Density and Expectation
1
EE 325 Probability and Random Processes
Oct 31,2013
Markovs Inequality
Recall the Markovs inequality for the discrete random variab
EE325 Tutorial III
1. Let Xn X w.p. 1 and Xn Y w.p. 1. Then show that X = Y w.p. 1.
2. Let Xn X weakly and Xn Y weakly. Then show that X = Y w.p. 1.
3. Let cfw_Xn n1 be monotone increasing sequence of the random variable such
that Xn X weakly. Show that X