STOCHASTIC PROCESSES:
Theory for Applications
Draft
R. G. Gallager
December 2, 2013
i
ii
Preface
This text has evolved over some 20 years, starting as lecture notes for two first-year graduate
subjects at M.I.T., namely, Discrete Stochastic Processes (6.2

Northwestern University
Department of Electrical Engineering and Computer Science
EECS 422
Winter 2014
Midterm Solutions:
1.
a.) FALSE: this is true for independent events but not disjoint events (unless one
has prob. 0).
b.) TRUE.
c.) FALSE: its expected

Northwestern University
Department of Electrical Engineering and Computer Science
EECS 422
Midterm Exam
Winter 2014
Date Issued: Feb. 13, 2014
You have 1 hour, 50 minutes to complete the exam.
The exam is closed book but you may use up to 2 8.5 11 pages

Northwestern University
Department of Electrical Engineering and Computer Science
EECS 422
Winter 2015
Problem Set 5 Solutions:
1.
a.) Using the linearity of expectation we have E(Z 2 ) = E(aX by)2 ) = a2 E(X 2 )
2abE(XY ) + b2 E(Y 2 ).
b.) First note th

Northwestern University
Department of Electrical Engineering and Computer Science
EECS 422
Winter 2015
Problem set 2 Solutions:
Problems:
1. We have
FY (y) = Pr(Y y)
= Pr(F 1 (X) y)
= Pr(X F (y)
= F (y)
3.
since F is invertible
Since X is uniform on [0, 1

Northwestern University
Department of Electrical Engineering and Computer Science
EECS 422
Winter 2015
Problem set 3 Solutions:
1.
a.) We show that Yn converges to 1 in probability. To see this note that for any
(0, 1]1
Pr(|Yn 1| > ) = Pr(cfw_Yn < 1 )
=

Northwestern University
Department of Electrical Engineering and Computer Science
EECS 422
Winter 2015
Problem Set 4 Solutions:
4.
a.) The transition matrix is given by
p
(1 p
P =
(1 p)
p
.
b.) We can find the stationary distribution q by solving
q = qP
w

Northwestern University
Department of Electrical Engineering and Computer Science
EECS 422
Winter 2015
Problem set 1 Solutions:
Problems:
1.
a.) The sample space can be written as S = cfw_HH, HT H, T HH, HT T H, T HT H, T T HH, . . .
where for example T H