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
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 sever
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 th
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 st
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 cha
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 ospr
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),
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. As
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 i
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
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 no
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
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 ( ) =
=
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