UNIVERSITY OF TORONTO
Department of Electrical and Computer Engineering
Random Processes-ECE537
Fall 2013
Homework #1 (Solutions)
1. Problem 2-6.
Series S consists of a countable number of elements, any subset of S is a countable union of
elementary event
UNIVERSITY OF TORONTO
Department of Electrical and Computer Engineering
Random Processes-ECE537
Fall 2013
Homework #2 (Solutions)
1. Problem 6-17.
(a) (See Equation 5.48)
We have independent Poisson random variables, therefore:
for k1 0, k2 0, k3 0
(1 t)k
ECE 537H1S - Random Processes, Fall 2014
Synopsis: Introduction to the principles and properties of random processes, with applications to
communications, control systems, and computer science.
Prerequisites: Introductory probability (ECE 302), linear sys
Homework #10
1. (Resnick, Adventures in Stochastic Processes) The Media Police have identied six states
associated with television watching: 0 (never watch TV), 1 (watch only PBS), 2 (watch TV
fairly frequently), 3 (addict), 4 (undergoing behavior modicat
Pairs of Random Variables
Random Processes for
Engineering Applications
Lecture 6
Vector Random Variables
A vector random variable is a function that
assigns a vector of real numbers to each
outcome in S.
X() = x
S
Rn
Select a students name from an urn,
Gaussian Vector Random
Variables
Joint pdf of Vector Random Variables
X1, X2, , Xn are jointly continuous if the probability
of any n-dimensional event A is given by an ndimensional integral of a pdf:
P (X1,., Xn ) in A = ! fX ,.,X (x1,., xn )dx1,.,dxn
x
Random Processes
Random Processes for
Engineering Applications
Lecture 15
Family of Random Variables
In many random experiments, the outcome
is a function of time or space.
Voltage waveform corresponding to speech
utterance
Number of customers in queue
Pairs of Random Variables
Random Processes for
Engineering Applications
Lecture 6
Vector Random Variables
A vector random variable is a function that
assigns a vector of real numbers to each
outcome in S.
X() = x
S
Rn
An event involving X = (X1, X2, , X
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of your last name in
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UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
ELECTRICAL AND COMPUTER ENGINEERING
ECE 537F Random Processes
Final Examination
2:00 pm - 4:30 pm, December
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UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
ELECTRICAL AND COMPUTER ENGINEERING
ECE 537F ' Random Processes
Final Examination
2:00 pm - 4:30 pm, December
Introduction and Course
Overview
Random Processes for
Engineering Applications
Randomness in ECE Systems
Variability in environment
Noise & interference in communications
Variability in Internet traffic
Incomplete control in system parameters
Wavelen
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of your last name in
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UNIVERSITY OF TORONTO
l FACULTY OF APPLIED SCIENCE AND ENGINEERING
i ELECTRICAL AND COMPUTER ENGINEERING
1 ECE 537F Probability and Random Processes
Final Examination
9:30 am
0
UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
- FINAL EXAMINATION - DECEMBER 2011
. ECE537F - RANDOM' PROCESSES
III',IV-AECPEBASC, III,IVAEELEBASC, IV-AEESCBASEF, IV-AEESCBASER
EXAMINER - R.H. Kwong
SURNAME
GIVEN NAME
STUDENT NUMBER
D
Sums of Random Variables
Random Processes for
Engineering Applications
Lecture 12
Sums of Random Variables
Let X1, X2, Xn be a sequence of random
variables, and let Sn be their sum:
Sn = X1 + X2 + + Xn
Sn is is sequence of random variables
What happens
Fine Points: Event Classes and
Probabilities of Sequences of
Events
Random Processes for
Engineering Applications
Lecture 3-4
Sections 2.8 and 2.9
Sample Space and Events
A
An outcome or sample point of a random
experiment is a result that cannot be
deco
Functions of a Random Variable
Random Processes for
Engineering Applications
Lecture 5
Functions of a Random Variable
Very often we are interested in a function of a
random variable: Y = g(X)
Since Y = g(X(), Y is also a random variable
Y = aX + b
Y =
Random Processes
Random Processes for
Engineering Applications
Lecture 15
Family of Random Variables
In many random experiments, the outcome
is a function of time or space.
Voltage waveform corresponding to speech
utterance
Number of customers in queue
Pairs of Random Variables
Random Processes for
Engineering Applications
Lecture 6
Vector Random Variables
A vector random variable is a function that
assigns a vector of real numbers to each
outcome in S.
X() = x
S
Rn
An event involving X = (X1, X2, , X
Sums of Random Variables
Random Processes for
Engineering Applications
Lecture 12
Sums of Random Variables
Let X1, X2, Xn be a sequence of random
variables, and let Sn be their sum:
Sn = X1 + X2 + + Xn
Sn is is sequence of random variables
What happens
Examples of Random Processes
Random Processes for
Engineering Applications
Lecture 18-19
Interarrival Times in Poisson Process
Find the pdf of the time T till the first arrival in a
Poisson process
P[T > t] = P[no events in t seconds] =
The interevent t
ECE 537H1S - Random Processes
Introduction to the principles and properties of random processes, with applications to communications,
control systems, and computer science. Topics include random vectors, random convergence, random
processes, specifying ra
Fine Points: Event Classes and
Probabilities of Sequences of
Events
Random Processes for
Engineering Applications
Lecture 3-4
Sections 2.8 and 2.9
Sample Space and Events
A
An outcome or sample point of a random
experiment is a result that cannot be
deco
Sums of Random Variables
Random Processes for
Engineering Applications
Lecture 12
Sums of Random Variables
Let X1, X2, Xn be a sequence of random
variables, and let Sn be their sum:
Sn = X1 + X2 + + Xn
Sn is is sequence of random variables
What happens
Mean-Square Calculus of
Random Processes
Random Processes for
Engineering Applications
Properties of Sample Functions of RPs
Are the sample functions continuous?
Do the sample functions have derivatives?
Can the sample functions be integrated?
Possibl
ECE 537 Midterm Exam
Open Book and Single Aid Sheet, 2 hours
Problem 1. (15 marks) Let X,Y,Z be independent exponentially-distributed random variables with
parameters A1, A2, 7L3. Define the event A=cfw_ x,y,z | min(x,z) > a, and y<b. Find:
a) P[XSX,YSy,Z
Sums of Random Variables
Random Processes for
Engineering Applications
Lecture 12
Sums of Random Variables
Let X1, X2, Xn be a sequence of random
variables, and let Sn be their sum:
Sn = X1 + X2 + + Xn
Sn is is sequence of random variables
What happens
Homework #1
(Probability review; problems from Papoulis & Pillai; note slightly different notations.)
2-6. Show that if S consists of a countable number of elements i and each subset cfw_i is an
event, then all subsets of S are events.
2-7. If S = cfw_1,