ECSE-305 Probability and Random Signals
Solutions to Midterm Examination II
8:35h - 9.55h. Wednesday, 19th March, 2014
Closed book examination.
Instructor: Peter E. Caines
No notes or calculators.
Begin each question on a new page.
Question 1 [20 points]

ECSE-305 Probability and Random Signals
Midterm Examination II
8:35h - 9.55h. Thursday, 24th March, 2016
Closed book examination.
Instructor: Peter E. Caines
No notes or calculators.
Begin each question on a new page.
Question 1 [20 points]
(a) The expone

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

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

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

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

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

ECSE-305 Probability and Random Signals
Midterm Examination I
8:35am - 9.55am. Thursday, 25th February, 2016
Prof. Peter E. Caines
Closed book examination. No notes or calculators permitted. You may keep the exam.
Give numerical answers as integers or sim

ECSE-305 Probability and Random Signals
Midterm Examination II
8:35h - 9.55h. Wednesday, 19th March, 2014
Closed book examination.
Instructor: Peter E. Caines
No notes or calculators.
Begin each question on a new page.
Question 1 [20 points]
Let X and Y b

ECSE-305 Probability and Random Signals
Midterm Examination II
9:35am - 10.25am. Friday, 18th November, 2016
Professor Peter E. Caines
Closed book examination. No notes or calculators permitted. Please keep the examination paper.
Give numerical answers as

ECSE-305 Probability and Random Signals
Midterm Examination I
9:35am - 10.25am. Friday, 14th October, 2016
Professor Peter E. Caines
Closed book examination. No notes or calculators permitted. You may keep the exam.
Give numerical answers as integers or s

ECSE-305 Probability and Random Signals
Midterm Examination II
8:35h - 9.55h. Thursday, 24th March, 2016
Closed book examination.
Instructor: Peter E. Caines
No notes or calculators.
Begin each question on a new page.
Question 1 [20 points]
(a) The expone

ECSE 305
Fall 2011
Midterm 2 - Solution Set
Question 1 [25]
Let X and Y be independent Gaussian random variables corresponding to the intensities of two
current sources with means , , respectively, and variances, 2 0, 2 0, respectively.
(a) Find the chara

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

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

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)

3. Multivariate random variables
3.1 Vector random variables, joint CDF and
statistical independence
Consider random variables 1 , 2 , , defined over a probability space , F, .
Let 1 , 2 , , be non-pathological subsets of .
Hence for = 1, 2, , we have 1 (

2. Scalar random variables
2.1 The concept of a random variable and the
Cummulative Distribution Function (CDF)
The sample space = 1, 2 , associated with a random experiment may be
composed of outcomes that are not necessary numerical.
For example, in a c

ECSE-305B Probability and Random Signals
Midterm Examination II
8:35h - 9.55h. Monday, 21st March, 2005
Closed book examination.
Instructor: Peter E. Caines
No notes or calculators.
Begin each question on a new page.
Question 1 [20 points]
Let X and Y be

ECSE-305 Probability and Random Signals
Midterm Examination I
8:35am - 9.55am. Thursday, 25th February, 2016
Prof. Peter E. Caines
Closed book examination. No notes or calculators permitted. You may keep the exam.
Give numerical answers as integers or sim

McGill University
Dept. of Electrical and Computer Eng.
ECSE-305B Probability and Random Signals I
Winter 2015
Term test # 2
Tuesday, March 17, 2015, 8:35 9:45 AM
Closed books and closed notes test. One crib sheet allowed (letter size, double sided printe