Solutions to Homework 1
6.262 Discrete Stochastics Process
MIT, Spring 2011
Solution to Exercise 1.3:
a) Since A1 , A2 , . . . , are assumed to be disjoint, the third axiom of probability says that
Am =
Pr
Pr Am
m=1
m=1
Since = m=1 Am , the term on the le
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering and Computer Science
6.262 Discrete Stochastic Processes
Midterm Quiz
April 6, 2010
There are 5 questions, each with several parts. If any part of any question is unclear to you
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
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
1
Solution to 6.262 Final Examination 5/21/2009
Solution to Question 1
a) We rst solve the steady state equations for the Markov process. As we have seen many
times, the pi , i 0 for a birth-death chain are related by pi+1 qi+1,i = pi qi,i+1 , which in th
Solutions to Homework 9
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 ospring of an individual.
a) Argue that limn Xn exists with
Solutions to Homework 8
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
Solutions to Homework 7
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
Solutions to Homework 6
6.262 Discrete Stochastic Processes
MIT, Spring 2011
Exercise 1
Let cfw_Yn ; n 1 be a sequence of rvs and assume that limn E[|Yn |] = 0. Show that
cfw_Yn ; n 1 converges to 0 in probability. Hint 1: Look for the easy way. Hint 2: T
Solutions to Homework 5
6.262 Discrete Stochastic Processes
MIT, Spring 2011
Solution to Exercise 2.28:
Suppose that the states are numbered so that state 1 to J1 are in the recurrent class 1,
J1 + 1 to J1 + J2 in recurrent class 2, etc. Thus, [P ] has th
Solutions to Homework 4
6.262 Discrete Stochastic Processes
MIT, Spring 2011
Solution to Exercise 2.28:
The purpose of this problem is to illustrate that for an arrival process with independent
but not identically distributed interarrival intervals, X1 ,
Solutions to Homework 3
6.262 Discrete Stochastic Processes
MIT, Spring 2011
Solution to Exercise 2.3:
a) Given Sn = , we see that N (t) = n, for t only if there are no arrivals from to
t. Thus,
Pr (N (t) = n|Sn = ) = exp (t )
b)
t
Pr (N (t) = n)|Sn = )fS
Solutions to Homework 2
6.262 Discrete Stochastic Processes
MIT, Spring 2011
Solution to Exercise 1.10:
a) We know that Z ( ) = X ( ) + Y ( ) for each .
Pr(Z ( ) = ) = Prcfw_ ; Z ( ) = + or Z ( ) =
= Prcfw_ ; Z ( ) = + + Prcfw_ ; Z ( ) =
Prcfw_ ; Z ( )
6.262 Discrete Stochastic Processes
MIT, Fall 2011
Monday April 4, 7:00-9:30pm, 2011
Solutions to Quiz
Problem 1: An innite sequence of packets are waiting to be sent, one after the other,
from point A to an intermediate point B and then on to C . The len