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  • B

    PROBABILITY ANDRANDOM PROCESSES

  • WILEY SURVIVAL GUIDES IN ENGINEERINGAND SCIENCE

    Emmanuel Desurvire, Editor

    Wiley Survival Guide in Global Telecommunications: Signaling Principles,

    Network Protocols, and Wireless Systems Emmanuel Desurvire

    Wiley Survival Guide in Global Telecommunications: Broadband Access,

    Optical Components and Networks, and CryptographyEmmanuel Desurvire

    Fiber to the Home: The New Empowerment Paul E. Green, Jr.

    Probability and Random Processes Venkatarama Krishnan

  • BPROBABILITY ANDRANDOM PROCESSES

    Venkatarama KrishnanProfessor Emeritus of Electrical Engineering

    University of Massachusetts Lowell

    A John Wiley & Sons, Inc., Publication

  • Copyright # 2006 by John Wiley & Sons, Inc. All rights reserved

    Published by John Wiley & Sons, Inc., Hoboken, New Jersey

    Published simultaneously in Canada

    No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form

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    Library of Congress Cataloging-in-Publication Data:

    Krishnan, Venkatarama,

    Probability and random processes / by Venkatarama Krishnan.p. cm. (Wiley survival guides in engineering and science)

    Includes bibliographical references and index.

    ISBN-13: 978-0-471-70354-9 (acid-free paper)

    ISBN-10: 0-471-70354-0 (acid-free paper)

    1. Probabilities. 2. Stochastic processes. 3. EngineeringStatistical methods.

    4. ScienceStatistical methods. I. Title. II. Series

    QA273.K74 2006

    519.2dc22 2005057726

    Printed in the United States of America

    10 9 8 7 6 5 4 3 2 1

  • Contents

    Preface, xi

    CHAPTER 1Sets, Fields, and Events, 1

    1.1 Set Definitions, 1

    1.2 Set Operations, 3

    1.3 Set Algebras, Fields, and Events, 8

    CHAPTER 2Probability Space and Axioms, 10

    2.1 Probability Space, 10

    2.2 Conditional Probability, 14

    2.3 Independence, 18

    2.4 Total Probability and Bayes

    Theorem, 20

    CHAPTER 3Basic Combinatorics, 25

    3.1 Basic Counting Principles, 25

    3.2 Permutations, 26

    3.3 Combinations, 28

    CHAPTER 4Discrete Distributions, 37

    4.1 Bernoulli Trials, 37

    4.2 Binomial Distribution, 38

    4.3 Multinomial Distribution, 41

    4.4 Geometric Distribution, 42

    4.5 Negative Binomial Distribution, 44

    4.6 Hypergeometric Distribution, 46

    4.7 Poisson Distribution, 48

    4.8 Logarithmic Distribution, 55

    4.9 Summary of Discrete Distributions, 62

    CHAPTER 5Random Variables, 64

    5.1 Definition of Random Variables, 64

    5.2 Determination of Distribution and

    Density Functions, 66

    5.3 Properties of Distribution and Density

    Functions, 73

    5.4 Distribution Functions from Density

    Functions, 75

    CHAPTER 6Continuous Random Variables and BasicDistributions, 79

    6.1 Introduction, 79

    6.2 Uniform Distribution, 79

    6.3 Exponential Distribution, 80

    6.4 Normal or Gaussian Distribution, 84

    CHAPTER 7Other Continuous Distributions, 95

    7.1 Introduction, 95

    7.2 Triangular Distribution, 95

    7.3 Laplace Distribution, 96

    7.4 Erlang Distribution, 97

    vii

  • 7.5 Gamma Distribution, 99

    7.6 Weibull Distribution, 101

    7.7 Chi-Square Distribution, 102

    7.8 Chi and Other Allied Distributions,

    104

    7.9 Student-t Density, 110

    7.10 Snedecor F Distribution, 111

    7.11 Lognormal Distribution, 112

    7.12 Beta Distribution, 114

    7.13 Cauchy Distribution, 115

    7.14 Pareto Distribution, 117

    7.15 Gibbs Distribution, 118

    7.16 Mixed Distributions, 118

    7.17 Summary of Distributions of

    Continuous Random Variables, 119

    CHAPTER 8Conditional Densities and Distributions, 122

    8.1 Conditional Distribution and Density

    for P(A)= 0, 122

    8.2 Conditional Distribution and Density

    for P(A) 0, 1268.3 Total Probability and Bayes Theorem

    for Densities, 131

    CHAPTER 9Joint Densities and Distributions, 135

    9.1 Joint Discrete Distribution

    Functions, 135

    9.2 Joint Continuous Distribution

    Functions, 136

    9.3 Bivariate Gaussian Distributions, 144

    CHAPTER 10Moments and Conditional Moments, 146

    10.1 Expectations, 146

    10.2 Variance, 149

    10.3 Means and Variances of Some

    Distributions, 150

    10.4 Higher-Order Moments, 153

    10.5 Bivariate Gaussian, 154

    CHAPTER 11Characteristic Functions and GeneratingFunctions, 155

    11.1 Characteristic Functions, 155

    11.2 Examples of Characteristic

    Functions, 157

    11.3 Generating Functions, 161

    11.4 Examples of Generating

    Functions, 162

    11.5 Moment Generating Functions, 164

    11.6 Cumulant Generating Functions, 167

    11.7 Table of Means and Variances, 170

    CHAPTER 12Functions of a Single Random Variable, 173

    12.1 Random Variable g(X ), 173

    12.2 Distribution of Y g(X ), 17412.3 Direct Determination of Density

    fY ( y) from fX(x), 194

    12.4 Inverse Problem: Finding g(x) Given

    fX(x) and fY ( y), 200

    12.5 Moments of a Function of

    a Random Variable, 202

    CHAPTER 13Functions of Multiple RandomVariables, 206

    13.1 Function of Two Random Variables,

    Z g(X,Y ), 20613.2 Two Functions of Two Random

    Variables, Z g(X,Y ),W h(X,Y ), 222

    13.3 Direct Determination of Joint Density

    fZW(z,w ) from fXY(x,y ), 227

    13.4 Solving Z g(X,Y ) Using an AuxiliaryRandom Variable, 233

    13.5 Multiple Functions of

    Random Variables, 238

    viii Contents

  • CHAPTER 14Inequalities, Convergences, and LimitTheorems, 241

    14.1 Degenerate Random Variables, 241

    14.2 Chebyshev and Allied

    Inequalities, 242

    14.3 Markov Inequality, 246

    14.4 Chernoff Bound, 248

    14.5 CauchySchwartz Inequality, 251

    14.6 Jensens Inequality, 254

    14.7 Convergence Concepts, 256

    14.8 Limit Theorems, 259

    CHAPTER 15Computer Methods for GeneratingRandom Variates, 264

    15.1 Uniform-Distribution Random

    Variates, 264

    15.2 Histograms, 266

    15.3 Inverse Transformation

    Techniques, 269

    15.4 Convolution Techniques, 279

    15.5 AcceptanceRejection

    Techniques, 280

    CHAPTER 16Elements of Matrix Algebra, 284

    16.1 Basic Theory of Matrices, 284

    16.2 Eigenvalues and

    Eigenvectors of Matrices, 293

    16.3 Vectors and Matrix

    Differentiations, 301

    16.4 Block Matrices, 308

    CHAPTER 17Random Vectors andMean-Square Estimation, 311

    17.1 Distributions and Densities, 311

    17.2 Moments of Random Vectors, 319

    17.3 Vector Gaussian Random

    Variables, 323

    17.4 Diagonalization of

    Covariance Matrices, 330

    17.5 Simultaneous Diagonalization of

    Covariance Matrices, 334

    17.6 Linear Estimation of Vector

    Variables, 337

    CHAPTER 18Estimation Theory, 340

    18.1 Criteria of Estimators, 340

    18.2 Estimation of Random

    Variables, 342

    18.3 Estimation of Parameters (Point

    Estimation), 350

    18.4 Interval Estimation (Confidence

    Intervals), 364

    18.5 Hypothesis Testing (Binary), 373

    18.6 Bayesian Estimation, 384

    CHAPTER 19Random Processes, 406

    19.1 Basic Definitions, 406

    19.2 Stationary Random Processes, 420

    19.3 Ergodic Processes, 439

    19.4 Estimation of Parameters of

    Random Processes, 445

    19.5 Power Spectral Density, 472

    CHAPTER 20Classification of Random Processes, 490

    20.1 Specifications of Random

    Processes, 490

    20.2 Poisson Process, 492

    20.3 Binomial Process, 505

    20.4 Independent Increment Process, 507

    20.5 Random-Walk Proce

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