Transformation of a RV
Introduction
X()
R
h()
R
0
S
1
1
1
s1
s2
s3
s4
2
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
39
Transformation of a RV
Introduction
Y = h(X)
S
R
1
s1
s2
s3
1
s4
ECSE 305 - Winter 2012 (slides based on the note

Moments of Random Variables
SG: 246 - 260
The expectation of a discrete random variable X with
discrete probability mass function cfw_pk ; k Z is given by
EX =
x k pk
k=
if cfw_<k<|xk |pk <
The expectation of a continuous random variable X with
probabili

Probability Density Functions
SG: Chapter 6
A (cumulative) distribution function
F : R [0, 1]
has a density f () : R R+ if
F (x) =
x
f (t)dt
x R
Example: Uniform Probability Density Function
SG: Chapter 7: pp 261 - 264
A uniform probability density on the

Unit II Random Variables
SG: Chapter 4; 139 - 158
A random variable X is a function X : S R which
assigns a real number X(s) to each outcome s S of an
experiment E.
The theory of random variables is concerned with the
probabilistic behaviour of experiment

5
Bernouilli and Markov Stochastic Processes
A Sequential Experiment (with Random Outcomes)
consists of a sequence of individual trials (which may be
independent or dependent with respect to each other).
(In other words, a sequential experiment is a discr

McGill University
Dept. of Electrical and Computer Eng.
ECSE-305A Probability and Random Signals I
Fall 2013
Term test # 1
Thursday, Oct. 10, 2013, 11:35 12:25
Closed books and closed notes test. Only faculty calculator is allowed.
Attempt all three probl

1.1 Set Theory
= () + ( )
Conditional probability
Random Vars
Common Random Vars
Continuous Random Vars
Common Random Vars
The moment generating function and the characteristic function
Multivariate random variables
Jacobean
Y=g(X), Then fY(y) =
(x)
Cond

Functions of a Random Variable
SG: 240 - 244; L-G: 119 - 126
Let X be a random variable and let y(.) be a
function y : R R. Let the random variable Y
be dened by Y = y(X).
Objective: nd the distribution (or density if it
exists) of Y given the distributio

The Central Limit Theorem
SG 498 - 505; L-G: 280 - 288
Let cfw_Xi, 1 i be a set of independent
identically distributed random variables each with
distribution function FX , mean value 0 and variance 2.
Set:
1
Sn
n
1
n
n
k=1 Xk ,
n Z+
Then:
n ()
S
= Ee

ECSE 305
Tutorial 1
Fall 2015
Tutorial 1 Questions
1. Consider the following two sets: A = cfw_a, b, c and B = cfw_c, d, e. You are given
that Ac B = cfw_c, d, e, f . What is B c equal to?
2. Let denote the set of real numbers = (, ).
(a) Use De Morgans l

ECSE 305
Tutorial 6
Fall 2015
Tutorial 6 Questions
1. Let X be a continuous RV with PDF
f (x) =
2|x|3 , x [1, 1]
0
otherwise
Using the method of transformations, nd g(y), the PDF of RV Y = 1 X 2 .
2. A production line manufactures 1000-ohm () resistors th

ECSE 305
Tutorial 4
Fall 2015
Tutorial 4 Questions
1. Let Y be an integer-valued random variable. Show that
P (Y = n) = P (Y > n 1) P (Y > n).
2. Let us construct a model for counting the number of heads in a sequence of three
coin tosses. For the underly

ECSE 305
Tutorial 2
Fall 2015
Tutorial 2 Questions
1. An urn contains 12 balls of 3 colours: 3 red, 4 blue, 5 green. Three balls are
taken from the urn at random without replacement. Find the probabilities that
the set of 3 balls will consist of:
(a) 1 ba

Transform Methods: Characteristic Functions
G: 457 - 476; L-G: 145 - 150
The characteristic function of a random variable
X is the Fourier transform of the (probability
law of the) distribution function FX (.) of X.
Explicitly:
X () =
jx
dFX (x)
e
= Eej

Unit IV
Stochastic Processes and Random Signals
SG Chapter 12
A stochastic process X is a time indexed set of
random variables X = cfw_X1, X2, ., Xn, . (discrete time)
or X = cfw_Xt; t R (continuous time).
Example: Independent Identically Distributed (IID

III
Multivariable or Vector Random Variables
RVs and Joint and Marginal Distributions
SG: Ch. 9
A vector random variable X is a function X : S Rn
which assigns a vector of real numbers X(s) Rn to each
outcome s S of an experiment E.
In particular, bivaria

Functions of Several Random Variables
Let X1, X2 be a vector random variable with cts.
probability density function fX1,X2 (x1, x2).
Let g(, ) be an R2 R2 one-to-one function (with
continuous derivatives) of the random variable (X1, X2)
into (Y1, Y2):
Y1

Problem set 5 : Text book pp. 170 - 178, problems: 2, 7, 12, 15, 23, 25, 28, 33
Set 6 : Text book pp. 222 - 236, problems: 2, 6, 7, 12, 32, 42, 44
Set 7 Text book pp. 319 - 326, problems: 12, 16, 17, 18, 19, 20
Set 8: Text book pp. 178 - 180, problems: 38

McGill University
Dept. of Electrical and Computer Eng.
ECSE-305A Probability and Random Signals I
Fall 2013
Term test # 2
Tuesday, Nov. 12, 2013, 11:35 12:25
Closed books and closed notes test. Only faculty calculator is allowed.
Attempt all three proble

Joint PMF
Denition
Let X and Y be discrete random variables with sets of possible values
RX = cfw_x1 , x2 , . and RY = cfw_y1 , y2 , ., respectively. We say that X
and Y are jointly discrete and we dene their joint probability mass
function (JPMF) as
p(x,

CHAPTERS 11 and 12
1
Multivariate expectations
2
Inequalities
3
Limit theorems
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
1
Multivariate expectations
X1 , . . . , Xn : RVs dened over the same sample space.
Y : transformation of X1

CHAPTERS 9 and 11
Bi-variate, and
Multi-variate distributions
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
1
Bivariate Distributions
Introduction
Example
Experiment: Pick a car out of a black Escort (Eb), a red Escort (Er),
a black M

CHAPTER 13
1
Introduction to Random Processes
2
Wide sense stationary processes
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
1
Random processes
Signal: Function of time (t)
In many cases, signals exhibit a random behavior.
Examples:

Random Variables
Classication of RVs
1
Discrete RVs
2
Continuous RVs
3
Mixed RVs
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
1
Chapter 8
MIXED RANDOM VARIABLES
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
2
Mix

Expected value
Denition
Let X be a continuous RV with PDF f (x). Provided the integral
|x|f (x)dx is nite, the expected value of X is dened as
E(X) =
x f (x) dx.
Notation: E(X), or x .
E(X): mean, expectation, or expected value.
ECSE 305 - Winter 2012 (s

Random Variables
Classication of RVs
1
Discrete RVs
2
Continuous RVs
3
Mixed RVs
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
1
Chapter 7
CONTINUOUS RANDOM VARIABLES
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)

Expectation
Final remarks
The mean is actually a characteristic of the PMF p(x):
E(X) =
xi p(xi )
xp(x) =
xRX
i
Some times we say A PMF with mean.
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
81
Variance
Motivation
We want measure a

Bivariate expectation
Let X and Y be two RVs dened over a common sample space, and
Z = h(X, Y )
where h : R2 R.
Question
What is E(Z)?
ECSE 305 - Winter 2012 (slides based on the notes by B. Champagne)
1
Bivariate expectation
Two cases:
X, Y : Discrete
Jo

ECSE 305 Probability and Random Signals
Solution Set
Question 1
Axiom I
OR
Axiom II
Axiom III
1
ECSE 305 Probability and Random Signals
Solution Set
Question 2
(i)
By Axiom III
=>
(ii)
B independent of C
=>
(iii)
(iv)
LHS :
RHS :
For LHS = RHS, we require

ECSE305B Probability and Random Signals I
Final Examination: Brief Solutions
9:00 am 12:00 pm
Question 1
April 22, 2005
Question 1 [25 points]
The scalar real valued wide sense stationary stochastic process y is generated by passing the
scalar white noise

ECSE-305 Probability and Random Signals
Midterm I Solutions
Question 1
19 February, Winter 2013
[15]
(i) Let P be a probability P : E(S) R on the set of events E(S) of a sample space S.
State the Axioms of Probability Theory for the probability triple (S,

ECSE-305 Probability and Random Signals
Midterm II Solutions
Question 1
4 April, Winter 2013
[15]
(a) Assuming it exists, define a density f for a random variable X.
(b) Define the convergence in probability of a sequence of random variables cfw_Xn , n =

ECSE305 Probability and Random Signals I
Final Examination
6:00 pm 9:00 pm
23rd April, 2013
Examiner: Professor Peter E. Caines Associate Examiner: Professor Aditya Mahajan
Attempt all 6 questions.
Question 1 [15 points]
Message packets arrive at a buffer