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# Livro Leon ProbStatistics

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Probability, Statistics, and Random Processes for Electrical Engineering Third Edition Alberto Leon-Garcia University of Toronto Upper Saddle River, NJ 07458 Library of Congress Cataloging-in-Publication Data Leon-Garcia, Alberto. Probability, statistics, and random processes for electrical engineering / Alberto Leon-Garcia. -- 3rd ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-13-147122-1 (alk. paper) 1. Electric engineering--Mathematics. 2. Probabilities. 3. Stochastic processes. I. Leon-Garcia, Alberto. Probability and random processes for electrical engineering. II. Title. TK153.L425 2007 519.202'46213--dc22 2007046492 Vice President and Editorial Director, ECS: Marcia J. Horton Associate Editor: Alice Dworkin Editorial Assistant: William Opaluch Senior Managing Editor: Scott Disanno Production Editor: Craig Little Art Director: Jayen Conte Cover Designer: Bruce Kenselaar Art Editor: Greg Dulles Manufacturing Manager: Alan Fischer Manufacturing Buyer: Lisa McDowell Marketing Manager: Tim Galligan © 2008 Pearson Education, Inc. Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice HallTM is a trademark of Pearson Education, Inc. MATLAB is a registered trademark of The Math Works, Inc. All other product or brand names are trademarks or registered trademarks of their respective holders. The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to the material contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of this material. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN 0-13-147122-8 978-0-13-147122-1 Pearson Education Ltd., London Pearson Education Australia Pty. Ltd., Sydney Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., Toronto Pearson Educación de Mexico, S.A. de C.V. Pearson Education\u2014Japan, Tokyo Pearson Education Malaysia, Pte. Ltd. Pearson Education, Upper Saddle River, New Jersey TO KAREN, CARLOS, MARISA, AND MICHAEL. This page intentionally left blank v Contents Preface ix CHAPTER 1 Probability Models in Electrical and Computer Engineering 1 1.1 Mathematical Models as Tools in Analysis and Design 2 1.2 Deterministic Models 4 1.3 Probability Models 4 1.4 A Detailed Example: A Packet Voice Transmission System 9 1.5 Other Examples 11 1.6 Overview of Book 16 Summary 17 Problems 18 CHAPTER 2 Basic Concepts of Probability Theory 21 2.1 Specifying Random Experiments 21 2.2 The Axioms of Probability 30 2.3 Computing Probabilities Using Counting Methods 41 2.4 Conditional Probability 47 2.5 Independence of Events 53 2.6 Sequential Experiments 59 2.7 Synthesizing Randomness: Random Number Generators 67 2.8 Fine Points: Event Classes 70 2.9 Fine Points: Probabilities of Sequences of Events 75 Summary 79 Problems 80 CHAPTER 3 Discrete Random Variables 96 3.1 The Notion of a Random Variable 96 3.2 Discrete Random Variables and Probability Mass Function 99 3.3 Expected Value and Moments of Discrete Random Variable 104 3.4 Conditional Probability Mass Function 111 3.5 Important Discrete Random Variables 115 3.6 Generation of Discrete Random Variables 127 Summary 129 Problems 130 * * * * vi Contents CHAPTER 4 One Random Variable 141 4.1 The Cumulative Distribution Function 141 4.2 The Probability Density Function 148 4.3 The Expected Value of X 155 4.4 Important Continuous Random Variables 163 4.5 Functions of a Random Variable 174 4.6 The Markov and Chebyshev Inequalities 181 4.7 Transform Methods 184 4.8 Basic Reliability Calculations 189 4.9 Computer Methods for Generating Random Variables 194 4.10 Entropy 202 Summary 213 Problems 215 CHAPTER 5 Pairs of Random Variables 233 5.1 Two Random Variables 233 5.2 Pairs of Discrete Random Variables 236 5.3 The Joint cdf of X and Y 242 5.4 The Joint pdf of Two Continuous Random Variables 248 5.5 Independence of Two Random Variables 254 5.6 Joint Moments and Expected Values of a Function of Two Random Variables 257 5.7 Conditional Probability and Conditional Expectation 261 5.8 Functions of Two Random Variables 271 5.9 Pairs of Jointly Gaussian Random Variables 278 5.10 Generating Independent Gaussian Random Variables 284 Summary 286 Problems 288 CHAPTER 6 Vector Random Variables 303 6.1 Vector Random Variables 303 6.2 Functions of Several Random Variables 309 6.3 Expected Values of Vector Random Variables 318 6.4 Jointly Gaussian Random Vectors 325 6.5 Estimation of Random Variables 332 6.6 Generating Correlated Vector Random Variables 342 Summary 346 Problems 348 * Contents vii CHAPTER 7 Sums of Random Variables and Long-Term Averages 359 7.1 Sums of Random Variables 360 7.2 The Sample Mean and the Laws of Large Numbers 365 Weak Law of Large Numbers 367 Strong Law of Large Numbers 368 7.3 The Central Limit Theorem 369 Central Limit Theorem 370 7.4 Convergence of Sequences of Random Variables 378 7.5 Long-Term Arrival Rates and Associated Averages 387 7.6 Calculating Distribution\u2019s Using the Discrete Fourier Transform 392 Summary 400 Problems 402 CHAPTER 8 Statistics 411 8.1 Samples and Sampling Distributions 411 8.2 Parameter Estimation 415 8.3 Maximum Likelihood Estimation 419 8.4 Confidence Intervals 430 8.5 Hypothesis Testing 441 8.6 Bayesian Decision Methods 455 8.7 Testing the Fit of a Distribution to Data 462 Summary 469 Problems 471 CHAPTER 9 Random Processes 487 9.1 Definition of a Random Process 488 9.2 Specifying a Random Process 491 9.3 Discrete-Time Processes: Sum Process, Binomial Counting Process, and Random Walk 498 9.4 Poisson and Associated Random Processes 507 9.5 Gaussian Random Processes, Wiener Process and Brownian Motion 514 9.6 Stationary Random Processes 518 9.7 Continuity, Derivatives, and Integrals of Random Processes 529 9.8 Time Averages of Random Processes and Ergodic Theorems 540 9.9 Fourier Series and Karhunen-Loeve Expansion 544 9.10 Generating Random Processes 550 Summary 554 Problems 557 * * * viii Contents CHAPTER 10 Analysis and Processing of Random Signals 577 10.1 Power Spectral Density 577 10.2 Response of Linear Systems to Random Signals 587 10.3 Bandlimited Random Processes 597 10.4 Optimum Linear Systems 605 10.5 The Kalman Filter 617 10.6 Estimating the Power Spectral Density 622 10.7 Numerical Techniques for Processing Random Signals 628 Summary 633 Problems 635 CHAPTER 11 Markov Chains 647 11.1 Markov Processes 647 11.2 Discrete-Time Markov Chains 650 11.3 Classes of States, Recurrence Properties, and Limiting Probabilities 660 11.4 Continuous-Time Markov Chains 673 11.5 Time-Reversed Markov Chains 686 11.6 Numerical Techniques for Markov Chains 692 Summary 700 Problems 702 CHAPTER 12 Introduction to Queueing Theory 713 12.1 The Elements of a Queueing System 714 12.2 Little\u2019s Formula 715 12.3 The M/M/1 Queue 718 12.4 Multi-Server Systems: M/M/c, M/M/c/c, And 727 12.5 Finite-Source Queueing Systems 734 12.6 M/G/1 Queueing Systems 738 12.7 M/G/1 Analysis Using Embedded Markov Chains 745 12.8 Burke\u2019s Theorem: Departures From M/M/c Systems 754 12.9 Networks of Queues: Jackson\u2019s Theorem 758 12.10 Simulation and Data Analysis of Queueing Systems 771 Summary 782 Problems