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
Chapter 4
RENEWAL PROCESSES
4.1
Introduction
Recall that a renewal process is an arrival process in which the interarrival intervals are
positive,1 independent and identically distributed (IID) random variables (rvs). Renewal
processes (since they are arr
Chapter 1
INTRODUCTION AND REVIEW
OF PROBABILITY
1.1
Probability models
Probability theory is a central eld of mathematics, widely applicable to scientic, techno
logical, and human situations involving uncertainty. The most obvious applications are to
sit
Chapter 5
COUNTABLE-STATE MARKOV
CHAINS
5.1
Introduction and classication of states
Markov chains with a countably-innite state space (more briey, countable-state Markov
chains) exhibit some types of behavior not possible for chains with a nite state spac
Appendix A
Table of standard random variables
The following tables summarize the properties of some common random variables. If a
density or PMF is specied only in a given region, it is assumed to be zero elsewhere. The
parameters , , and a are assumed to
6.262 Discrete Stochastic Processes
LEC #
TOPICS
1
Introduction and probability review
2
More review; the Bernoulli process
3
Laws of large numbers, convergence
4
Poisson (the perfect arrival process)
5
Poisson combining and splitting
6
From Poisson to Ma
6.262 Discrete Stochastic Processes Wednesday, Feb. 2, 2011
MIT, Spring 2011
Due: Wednesday, Feb. 9
Problem Set 1
Exercise 1.3. Let 1511,1512. . . . . be a sequence of disjoint events and assume that PHAR} :
2"”‘1 for each 1':- 2 1. Assume also that Q
DISCRETE STOCHASTIC PROCESSES
Draft of 2nd Edition
R. G. Gallager
January 31, 2011
i
ii
Preface
These notes are a draft of a major rewrite of a text [9] of the same name. The notes and the
text are outgrowths of lecture notes developed over some 20 years