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Introduction to Probability Models (10th Edition) Ross
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Introduction to Probability Models (Tenth Edition) Sheldon M. Ross University of Southern California Los Angeles, California 概率(随机过程)的经典著作
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Introduction to Probability
Models
Tenth Edition
Sheldon M. Ross
University of Southern California
Los Angeles, California
AMSTERDAM
•
BOSTON
•
HEIDELBERG
•
LONDON
NEW YORK
•
OXFORD
•
PARIS
•
SAN DIEGO
SAN FRANCISCO
•
SINGAPORE
•
SYDNEY
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TOKYO
Academic Press is an Imprint of Elsevier
Academic Press is an imprint of Elsevier
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Copyright © 2010 Elsevier Inc. All rights reserved.
No part of this publication may be reproduced or transmitted in any form or by any means, electronic or
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permission in writing from the publisher. Details on how to seek permission, further information about the
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Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions.
This book and the individual contributions contained in it are protected under copyright by the Publisher
(other than as may be noted herein).
Notices
Knowledge and best practice in this field are constantly changing. As new research and experience broaden
our understanding, changes in research methods, professional practices, or medical treatment may become
necessary.
Practitioners and researchers must always rely on their own experience and knowledge in evaluating and
using any information, methods, compounds, or experiments described herein. In using such information
or methods they should be mindful of their own safety and the safety of others, including parties for whom
they have a professional responsibility.
To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any
liability for any injury and/or damage to persons or property as a matter of products liability, negligence
or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in
the material herein.
Library of Congress Cataloging-in-Publication Data
Ross, Sheldon M.
Introduction to probability models / Sheldon M. Ross. – 10th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-12-375686-2 (hardcover : alk. paper) 1. Probabilities. I. Title.
QA273.R84 2010
519.2–dc22
2009040399
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ISBN: 978-0-12-375686-2
For information on all Academic Press publications
visit our Web site at www.elsevierdirect.com
Typeset by: diacriTech, India
Printed in the United States of America
091011 987654321
Contents
Preface xi
1 Introduction to Probability Theory 1
1.1 Introduction 1
1.2 Sample Space and Events 1
1.3 Probabilities Defined on Events 4
1.4 Conditional Probabilities 7
1.5 Independent Events 10
1.6 Bayes’ Formula 12
Exercises 15
References 20
2 Random Variables 21
2.1 Random Variables 21
2.2 Discrete Random Variables 25
2.2.1 The Bernoulli Random Variable 26
2.2.2 The Binomial Random Variable 27
2.2.3 The Geometric Random Variable 29
2.2.4 The Poisson Random Variable 30
2.3 Continuous Random Variables 31
2.3.1 The Uniform Random Variable 32
2.3.2 Exponential Random Variables 34
2.3.3 Gamma Random Variables 34
2.3.4 Normal Random Variables 34
2.4 Expectation of a Random Variable 36
2.4.1 The Discrete Case 36
2.4.2 The Continuous Case 38
2.4.3 Expectation of a Function of a Random Variable 40
2.5 Jointly Distributed Random Variables 44
2.5.1 Joint Distribution Functions 44
2.5.2 Independent Random Variables 48
2.5.3 Covariance and Variance of Sums of Random Variables 50
vi Contents
2.5.4 Joint Probability Distribution of Functions of Random
Variables 59
2.6 Moment Generating Functions 62
2.6.1 The Joint Distribution of the Sample Mean and Sample
Variance from a Normal Population 71
2.7 The Distribution of the Number of Events that Occur 74
2.8 Limit Theorems 77
2.9 Stochastic Processes 84
Exercises 86
References 95
3 Conditional Probability and Conditional Expectation 97
3.1 Introduction 97
3.2 The Discrete Case 97
3.3 The Continuous Case 102
3.4 Computing Expectations by Conditioning 106
3.4.1 Computing Variances by Conditioning 117
3.5 Computing Probabilities by Conditioning 122
3.6 Some Applications 140
3.6.1 A List Model 140
3.6.2 A Random Graph 141
3.6.3 Uniform Priors, Polya’s Urn Model, and Bose–Einstein
Statistics 149
3.6.4 Mean Time for Patterns 153
3.6.5 The k-Record Values of Discrete Random Variables 157
3.6.6 Left Skip Free Random Walks 160
3.7 An Identity for Compound Random Variables 166
3.7.1 Poisson Compounding Distribution 169
3.7.2 Binomial Compounding Distribution 171
3.7.3 A Compounding Distribution Related to the Negative
Binomial 172
Exercises 173
4 Markov Chains 191
4.1 Introduction 191
4.2 Chapman–Kolmogorov Equations 195
4.3 Classification of States 204
4.4 Limiting Probabilities 214
4.5 Some Applications 230
4.5.1 The Gambler’s Ruin Problem 230
4.5.2 A Model for Algorithmic Efficiency 234
4.5.3 Using a Random Walk to Analyze a Probabilistic
Algorithm for the Satisfiability Problem 237
4.6 Mean Time Spent in Transient States 243
4.7 Branching Processes 245
Contents vii
4.8 Time Reversible Markov Chains 249
4.9 Markov Chain Monte Carlo Methods 260
4.10 Markov Decision Processes 265
4.11 Hidden Markov Chains 269
4.11.1 Predicting the States 273
Exercises 275
References 290
5 The Exponential Distribution and the Poisson Process 291
5.1 Introduction 291
5.2 The Exponential Distribution 292
5.2.1 Definition 292
5.2.2 Properties of the Exponential Distribution 294
5.2.3 Further Properties of the Exponential Distribution 301
5.2.4 Convolutions of Exponential Random Variables 308
5.3 The Poisson Process 312
5.3.1 Counting Processes 312
5.3.2 Definition of the Poisson Process 313
5.3.3 Interarrival and Waiting Time Distributions 316
5.3.4 Further Properties of Poisson Processes 319
5.3.5 Conditional Distribution of the Arrival Times 325
5.3.6 Estimating Software Reliability 336
5.4 Generalizations of the Poisson Process 339
5.4.1 Nonhomogeneous Poisson Process 339
5.4.2 Compound Poisson Process 346
5.4.3 Conditional or Mixed Poisson Processes 351
Exercises 354
References 370
6 Continuous-Time Markov Chains 371
6.1 Introduction 371
6.2 Continuous-Time Markov Chains 372
6.3 Birth and Death Processes 374
6.4 The Transition Probability Function P
ij
(t) 381
6.5 Limiting Probabilities 390
6.6 Time Reversibility 397
6.7 Uniformization 406
6.8 Computing the Transition Probabilities 409
Exercises 412
References 419
7 Renewal Theory and Its Applications 421
7.1 Introduction 421
7.2 Distribution of N(t) 423
7.3 Limit Theorems and Their Applications 427
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