Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Exercise 4.10: Consider a variation of an M/G/1 queueing system in which there is no
facility to save waiting customers. Assume customers arrive according to a Poisson process
of ra
Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Exercise 4.11
a From the gure, conditional on Sn = t
s (i.e., conditional on the age at time t being
s and on N (t) = n), the probability that the next arrival occurs after time t +
6.262 Discrete Stochastic Processes
MIT, Spring 2011
Wednesday, May 18, 9:00-12:00 noon, 2011
Solutions to nal examination
Problem 1: A nal exam is started at time 0 for a class of n students. Each student
is allowed to work until completing the exam. It
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2007
Midterm exam, 7-9pm, (120 mins/70 pts)
6.436J/15.085J
10/23/07
Possibly useful facts:
(a) If X is uniform on [a, b], then the variance of X is (b a)2 /12.
(b) n=1 1/n is innite when 1, and nite when > 1.
Pro
Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Exercise 1.22:
For each of the following random variables, nd the interval (r , r+ ) over which the moment generating function g(r) exists. Determine in each case whether gX (r) exi
Solutions to Homework 1
6.262 Discrete Stochastic Processes
MIT Spring 2011
Exercise 6.5:
Consider the Markov process illustrated below. The transitions are labelled by the rate
qij at which those transitions occur. The process can be viewed as a single s
6.262 Discrete Stochastic Processes
MIT, Spring 2011
May 6, 2011, 2011
Solutions to practice problem set 12
Note: There is a minor error in the statement of Exercise 7.21, part b. The last equation
of that part should be ZJ = ( 2)n (2n 1)/(n2 n). The erro
Solutions to Homework
6.262 Discrete Stochastic Processes
MIT Spring 2011
Exercise 5.6:
Let cfw_Xn ; n
0 be a branching process with X0 = 1. Let Y , 2 be the mean and
variance of the number of o spring of an individual.
a Argue that limn!1 Xn exists with
1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.262 Discrete Stochastic Processes
Midterm Exam - Solutions
April 7, 2009
Problem 1
1a) (i) Recall that two states i and j in a Markov chain communicate if
M SS CHUSETTS INSTITUTE OF TECHNOLOGY
Fall 2008
Midterm exam, 7-9pm (120 mins/100 pts)
6.436J/15.085J
10/21/08
Problem 1: (15 points)
Let cfw_Xn be a sequence of random variables (i.e., measurable functions) dened
on the same probability space ( , F, ).