Probability,_Random_Variables_and_Stochastic

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PROBABILITY, RANDOM VARIABLES, AND STOCHASTIC PROCESSES FOURTH EDITION Athanasios Papoulis < University Professor Polytechnic University s. Unnikrishna Pillai Professor of Electrical and Computer Engineering Polytechnic University Boston Burr Ridge, IL Dubuque, IA Madison, WI N~w York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Mila!) Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

McGraw-Hill Higher ~~~~ Z!2 A Division 0{ The McGrAw-Hill Companies PROBABIUTY. RANDOM VARIABLES, AND STOCHASTIC PROCESSES. FOUR11:l EDmoN Published by McGraw-Hill, a business unit of The McGraw-Hili Companies, Inc •• 1221 Avenue of the Americas, New York. NY 10020. Copyright e 2002. 1991, 1984. 1965 by ne McGraw-Hill Companies, Inc. All rights reserved. No part of this publication may be reproduced or dislributed in any form or by any means, or stored in a database or retneval system, without the prior written consent of The McGraw-Hili Companies, Inc .. including, but not limited to. in any network or other electronic storage or transmission. 01 broadcast for distance learning. Some ancillaries. including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. International1234567890 QPFJQPF 09876543210 DomestiC! 1234567890 QPP/QPF 09876543210 ISBN 0-07-366011-6 ISBN 0-07-112256-7 (ISE) General manager: Thomas E. CAsson Publisher: Elizabeth A. JOI1U Sponsoring editor: Cotherine Fields Shultz Developmental editor: Michelle 1.. Flornenhoft Executive marketing manager: John Wannemacher Project manager: Sheila M. Frank Production supervisor: Sherry 1.. Kane Coordinator of freelance design: Rick D. Noel Cover designer: So Yon Kim Cover image: CPhotoDisc. Signature &rlu, Dice. SS1OO74 Supplement producer: Brenda A. Emzen Media technology senior producer: PhiUip Meek Compositor: Interactive Composition Corporation 1YPeface: /0/12 7imes Roman Printer: Quebecor World Fairfield. PA Library of Congress Cataloging-ln.PubJication Data Papoulis. Atbanasios. 1921- Probability, random variables. and stochastic processes I Atbanasios Papoulis. S. Unnikrishna PlIlai. - 4th ed. p.em. Includes bibliographical references and index. ISBN 0-07-366011-6 - 1. Probabilities. 2. Random variables. 3. Stochastic processes. l. Pillai. S. U~bna, 1955 -. ISBN 0-07-112256-7 (ISE) II. TIde. QA273 .P2 5 19.2---dc21 2002 2001044139 CIP INTERNATIONAL EDmON ISBN 0-07-112256-7 Copyright C 2002. Exclusive rights by The McGraw-Hill Companies, Inc .. for manufacture and export. This book cannot be re-exported from the country to which it is sold by McGraw-Hut. The International Bdition is not available in North America. www.mhhe.com

CONTENTS Preface PART I PROBABILITY AND RANDOM VARIABLES Chapter! The Meaning of Probability 1-1 Introduction I 1-2 The Definitions I 1-3 Probability and Induction I 1-4 Causality Versus Randomness Chapter 2 The Axioms of Probability 2-1 Set Theory I 2-2 Probability Space I 2-3 Conditional Probability I Problems Chapter 3 Repeated Trials 3-1 Combined Experiments I 3-2 Bernoulli Trials I 3-3 Bernoulli's Theorem and Games of Chance I Problems Chapter 4 The Concept of a Random Variable 4-1 Introduction I 4-2 Distribution and Density Functions I 4-3 Specific Random Variables I 4-4 Conditional Distributions I 4-5 Asymptotic Approximations for Binomial Random Variable I Problems ChapterS Functions of One Random Variable 5-1 The Random Variable g(x) I 5-2 The Distribution " of g(x) I 5-3 Mean and Variance I 5-4 Moments I 5-5 Characteristic Functions I Problems Chapter 6 Two Random Variables 6-1 Bivariate Distributions I 6-2 One Function of Two Random Variables I 6-3 Two Functions of Two Random Variables I 6-4 Joint Moments I 6-5 Joint Characteristic Functions I 6-6 Conditional Distributions I 6-7 Conditional Expected Values I Problems ix 1 3 15 46 72 123 169

vi CONTENTS Chapter 7 Sequences of Random 'Variables 7-1 General Concepts / 7-2 Conditional Densities, Characteristic Functions, and Normality I 7-3 M~ Square Estimation I 7-4 Stochastic Convergence and Limit Theorems I 7-5 Random Numbers: Meaning and Generation I Problems Chapter 8 Statistics 8-1 Introduction I 8-2 Estimation I 8-3 Parameter Estimation I 8-4 Hypothesis Testing I Problems PART II STOCHASTIC PROCESSES Chapter 9 General Concepts 9-1 Definitions I 9-2 Systems with Stochastic Inputs I 9-3 The Power Spectrum I 9-4 Discrete-Time Processes I Appendix 9A Continuity, Differentiation, Integration I Appendix 9B Shift Operators and Stationary Processes I Problems Chapter 10 Random Walks and Other Applications 10-1 Random Walks I 10-2 Poisson Points and Shot Noise I 10-3 Modulation I 10-4 Cyclostationary Processes I 10-5 Bandlimited Processes and Sampling Theory I 10-6 Deterministic Signals in Noise I 10-7 Bispectra and System Identification I Appendix lOA The Poisson Sum Formula I Appendix lOB The Schwarz Inequality I Problems Chapter 11 Spectral Representation 11-1 Factorization and Innovations I 11-2 Finite-Order Systems and State Variables I 11-3 Fourier Series and Karhunen-Loeve Expansions I 11-4 Spectral Representation of Random Processes I Problems Chapter 12 Spectrum Estimation 12-1 Ergodicity I 12-2 Spectrum Estimation I 12-3 Extrapolation and System Identification I 12-4 The GeQeral Class of Extrapolating Spectra and Youla's Parametrization I Appendix 12A Minimum-Phase Functions I Appendix 12B All-Pass Functions I Problems Chapter 13 Mean Square Estimation 13-1 Introduction I 13-2 Prediction I 13-3 Filtering and Prediction I 13-4 Kalman Filters I Problems Chapter 14 Entropy 14-1 Introduction I 14-2 Basic Concepts I 14-3 Random Variables and Stochastic Processes I 14-4 The Maximum Entropy Method I 14-5 Coding I 14-6 Channel Capacity I Problems 243 303 371 373 435 499 523 580 629

Chapter 15 Markov Chains CONTENTS vii 695 15-1 InlI'Oduction I 15-2 Higher Transition Probabilities and the Chapman-Kolmogorov Equation I 15-3 Classification of StaleS I 15-4 Stationary Distributions and Limiting Probabilities I IS-S Transient States and Absorption Probabilities I 15-6 Branching Processes I Appendix 15A Mixed Type Population of Constant Size I Appendix. ISB Structure of Periodic Chains I Problems Chapter 16 Markov Processes and Queueing Theory 16-1 Introduction I 16-2 Markov Processes I 16-3 Queueing Theory I 16-4 Networks of Queues I Problems Bibliography Index 773 835 837

PREFACE The fourth edition of this book has been updated significantly from previous editions. arid it includes a coauthor. About one-third of the content of this edition is new material, and these additions are incorporated while maintaining the style and spirit of the previous editions that are familiar to many of its readers. The basic outlook and approach remain the same: To develop the subject of proba bility theory and stochastic processes as a deductive discipline and to illustrate the theory with basic applications of engineeling interest. To this extent. these remarks made in the first edition are still valid: "The book is written neither for the handbook-oriented stu dents nor for the sophisticated few (if any) who can learn the subject from advanced mathematical texts. It is written for the majority of engineers and physicists who have sufficient maturity to appreciate and follow a logical presentation .... There is an obvi ous lack of continuity between the elements of probability as presented in introductory courses, and the sophisticated concepts needed in today's applications .... Random vari ables. transformations, expected values, conditional densities, characteristic functions cannot be mastered with mere exposure. These concepts must be clearly defined and must be developed, one at a time, with sufficient elaboration." Recognizing these factors, additional examples are added for further clarity, and the new topics include the following. Chapters 3 and 4 have ul)dergone substantial rewriting. Chapter 3 has a detailed section on Bernoulli's theorem and games of chance (Sec. 3-3), and several examples are presented there including the classical gambler's ruin problem to stimulate student interest. In Chap. 4 various probability distributions are categorized and illustrated, and two kinds of approximations to the binomial distribution are carried out to illustrate the connections among some of the random variables. " Chapter 5 contains new examples illustrating the usefulness of characteristic func tions and moment-generating functions including the proof of the DeMoivre-Laplace theorem. Chapter 6 has been rewritten with additional examples, and is complete in its description of two random variables and their properties. Chapter 8 contains a new Sec. 8-3 on Parameter e6Eimation that includes key ideas on minimum variance unbiased estimation, the Cramer-Rao bound, the Rao-Blackwell theorem, and the Bhattacharya bound.

PREFACE . In Chaps. 9 and la, sections on Poisson processes are further expanded with additional results. A new detailed section on random walks has also been added. Chapter 12 includes a new subsection describing the parametrization of the class of all admissible spectral extensions given a set of valid autocorrelations. Because of the importance of queueing theory, the old material has undergone com plete revision to the extent that two new chapters (15 and 16) are devoted to this topic. Chapter 15 describes Markov chains, their properties, characterization, and the long-term (steady state) and transient behavior of the chain and illustrates various theorems through several examples. In particular, Example 15-26 The Game of Tennis is an excellent illustration of the theory to analyze practical applications, and the chapter concludes with a detailed study of branching processes, which have important applications in queue ing theory. Chapter 16 describes Markov processes and queueing theory starting with the Chapman-Kolmogorov equations and concentrating on the birth-death processes to illustrate markovian queues. The treatment, however, includes non-markovian queues and machine servicing problems, and concludes with an introduction to the network of queues. The material in this book can be organized for various one semester courses: • Chapters 1 to 6: Probability Theory (for senior andlor first-level graduate students) • Chapters 7 and 8: Statistics and Estimation Theory (as a follow-up course to Proba bility Theory) • Chapters 9 to 11: Stochastic Processes (follow-up course to Probability Theory.) • Chapters 12 to 14: Spectrum Estimation and Filtering (follow-up course to Stochastic Processes) • Chapters 15 and 16: Markov Chains and Queueing Theory (follow-up course to Probability Theory) The authors would like to thank Ms. Catherine Fields Shultz, editor for electrical and computer engineering at McGraw-Hill Publishing Company, Ms. Michelle Flomen hoft and Mr. John Griffin, developmental editors, Ms. Sheila Frank, Project manager and her highly efficient team, and Profs. D. P. Gelopulos, M. Georgiopoulos, A. Haddad, T. Moon, 1. Rowland, C. S. Tsang, J. K. Tugnait, and O. C. Ugweje, for their comments, criticism, and guidance throughout the period of this revision. In addition, Dr. Michael Rosse, several colleagues at Polytechnic including Profs. Dante Youla, Henry Bertoni, Leonard Shaw and Ivan Selesnick, as well as students Dr. Hyun Seok Oh. Mr. Jun Ho Jo. and Mr. Seung Hun Cha deserve special credit for their valuable help and encouragement during the preparation of the manuscript. Discussions with Prof. C. Radhakrishna Rao about two of his key theorems in statistics and other items are also gratefully acknowl edged. Athanasios PapouIis S. Unnikrishna Pillai

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