Introduction to Probability Dimitri MIT.pdf

Introduction to Probability

SECOND EDITION

Dimitri P. Bertsekas and John N. Tsitsiklis

Massachusetts Institute of Technology

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Publisher's Cataloging-in-Publication Data

Bertsekas, Dimitri P., Tsitsiklis, John N.

Introduction to Probability

Includes bibliographical references and index

Probabilities. 2. Stochstic Processes. I. Title.

QA273.B475 2008 519.2 - 21

Library of Congress Control Number: 2002092167

ISBN

978-1-886529-23-6

To the memory of

Pantelis Bertsekas and Nikos Tsitsiklis

Preface

Probab ility is common sense educed to calculation

Laplace

This book is an outgrowth of our involvement in teaching an introductory prob

ability course ("Probabilistic Systems Analysis'�) at the Massachusetts Institute

of Technology.

The course is attended by a large number of students with diverse back

grounds, and a broad range of interests. They span the entire spectrum from

freshmen to beginning graduate students, and from the engineering school to the

school of management. Accordingly, we have tried to strike a balance between

simplicity in exposition and sophistication in analytical reasoning. Our key aim

hs been to develop the ability to construct and analyze probabilistic models in

a manner that combines intuitive understanding and mathematical precision.

In this spirit, some of the more mathematically rigorous analysis hs been

just sketched or intuitively explained in the text. so that complex proofs do not

stand in the way of an otherwise simple exposition. At the same time, some of

this analysis is developed (at the level of advanced calculus) in theoretical prob

lems, that are included at the end of the corresponding chapter. Urthermore,

some of the subtler mathematical issues are hinted at in footnotes addressed to

the more attentive reader.

The book covers the fundamentals of probability theory (probabilistic mod

els, discrete and continuous random variables, multiple random variables, and

limit theorems), which are typically part of a irst course on the subject. It

also contains, in Chapters 4-6 a number of more advanced topics, from which an

instructor can choose to match the goals of a particular course. In particular, in

Chapter 4, we develop transforms, a more advanced view of conditioning, sums

of random variables, least squares estimation, and the bivariate normal distribu-

Preface

tion. Furthermore, in Chapters 5 and 6, we provide a fairly detailed introduction

to Bernoulli, Poisson, and Markov processes.

Our M.LT. course covers all seven chapters in a single semester, with the ex

ception of the material on the bivariate normal (Section 4.7), and on continuous

time Markov chains (Section 6.5). However, in an alternative course, the material

on stochstic processes could be omitted, thereby allowing additional emphsis

on foundational material, or coverage of other topics of the instructor's choice.

Our most notable omission in coverage is an introduction to statistics.

While we develop all the bsic elements of Bayesian statistics, in the form of

Bayes' rule for discrete and continuous models, and lest squares estimation, we

do not enter the subjects of parameter estimation, or non-Bayesian hypothesis

testing.

The problems that supplement the main text are divided in three categories:

(a) Theoretical poblems: The theoretical problems (marked by

) constitute

an important component of the text, and ensure that the mathematically

oriented reader will ind here a smooth development without major gaps.

Their solutions are given in the text, but an ambitious reader may be able

to solve many of them, especially in earlier chapters, before looking at the

solutions.

(b) Poblems in the text: Besides theoretical problems, the text contains several

problems, of various levels of diiculty. These are representative of the

problems that are usually covered in recitation and tutorial sessions at

M.LT., and are a primary mechanism through which many of our students

learn the material. Our hope is that students elsewhere will attempt to

solve these problems, and then refer to their solutions to calibrate and

enhance their understanding of the material. The solutions are posted on

the book's www site

http://www.athensc.com/probbook.html

ditional problems, which is not included in the book, but is made available

at the book's www site. Many of these problems have been ssigned s

homework or exam problems at M.I.T., and we expect that instructors

elsewhere will use them for a similar purpose. While the statements of

these additional problems are publicly accessible, the solutions are made

available from the authors only to course instructors.

We would like to acknowledge our debt to several people who contributed

in various ways to the book. Our writing project began when we ssumed re

sponsibility for a popular probability clss at M.LT. that our colleague Al Drake

had taught for several decades. We were thus fortunate to start with an organi

zation of the subject that had stood the test of time, a lively presentation of the

various topics in AI ' s clssic textbook, and a rich set of material that had been

used in recitation sessions and for homework. We are thus indebted to Al Drake

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