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Probability and Stochastic Processes

Features of this Text Who will beneﬁt from using this text? What’s New? This text can be used in Junior, Senior or graduate level courses in probability, stochastic process, random signal processing and queuing theory. The mathematical exposition will appeal to students and practioners in many areas. The examples, quizzes, and problems are typical of those encountered by practicing electrical and computer engineers. Professionals in the telecommunications and wireless industry will ﬁnd it particularly useful. This text has been expanded greatly with new material: • Matlab examples and problems give students hands-on access to theory and ap- plications. Every chapter includes guidance on how to use MATLAB to perform calculations and simulations relevant to the subject of the chapter. • A new chapter on Random Vectors • Expanded and enhanced coverage of Random Signal Processing • Streamlined exposition of Markov Chains and Queuing Theory provides quicker access to theories of greatest practical importance Notable Features The Friendly Approach The friendly and accessible writing style gives students an intuitive feeling for the formal mathematics. Quizzes and Homework Problems An extensive collection of in-chapter Quizzes provides check points for readers to gauge their understanding. Hundreds of end-of-chapter problems are clearly marked as to their degree of difﬁculty from beginner to expert. Website for Students http://www.wiley.com/college/yates Available for download: All Matlab m-ﬁles in the text, the Quiz Solutions Manual Instructor Support Instructors should register at the Instructor Companion Site (ISC) at Wiley in order to obtain supplements. The ISC can be reached by accessing the text’s companion web page http://www.wiley.com/college/yates • Unparalleled in its offerings, this Second Edition provides a web-based interface for instructors to create customized solutions documents that output in PDF or PostScript. • Extensive PowerPoint slides are available.

Probability and Stochastic Processes A Friendly Introduction for Electrical and Computer Engineers Second Edition Roy D. Yates Rutgers, The State University of New Jersey David J. Goodman Polytechnic University JOHN WILEY & SONS, INC.

EXECUTIVE EDITOR MARKETING MANAGER PRODUCTION EDITOR COVER DESIGNER Bill Zobrist Jennifer Powers Ken Santor Dawn Stanley This book was set in Times Roman by the authors using LATEXand printed and bound by Malloy, Inc. The cover was printed by Lehigh Press. About the cover: The cover shows a cutaway view of a bivariate Gaussian probability den- sity function. The bell-shaped cross-sections show that the marginal densities are Gaussian. This book is printed on acid-free paper. ∞ Copyright c 2005 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978)750-8400, fax (978)750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030 (201)748-6011, fax (201)748-6008, E-Mail: PERMREQ@WILEY.COM. To order books or for customer service call 1-800-CALL WILEY (225-5945). ISBN 0-471-27214-0 WIE 0-471-45259-9 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

To our children, Tony, Brett, and Zachary Yates Leila and Alissa Goodman

Preface What’s new in the second edition? We are happy to introduce you to the second edition of our textbook. Students and instructors using the ﬁrst edition have responded favorably to the “friendly” approach that couples engineering intuition to mathematical principles. They are especially pleased with the abundance of exercises in the form of “examples,” “quizzes,” and “problems,” many of them very simple. The exercises help students absorb the new material in each chapter and gauge their grasp of it. Aiming for basic insight, the ﬁrst edition avoided exercises that require complex com- putation. Although most of the original exercises have evident engineering relevance, they are considerably simpler than their real-world counterparts. This second edition adds a large set of Matlab programs offering students hands-on experience with simulations and calculations. Matlab bridges the gap between the computationally simple exercises and the more complex tasks encountered by engineering professionals. The Matlab section at the end of each chapter presents programs we have written and also guides students to write their own programs. Retaining the friendly character of the ﬁrst edition, we have incorporated into this edition the suggestions of many instructors and students. In addition to the Matlab programs, new material includes a presentation of multiple random variables in vector notation. This format makes the math easier to grasp and provides a convenient stepping stone to the chapter on stochastic processes, which in turn leads to an expanded treatment of the application of probability theory to digital signal processing. Why did we write the book? When we started teaching the course Probability and Stochastic Processes to Rutgers un- dergraduates in 1991, we never dreamed we would write a textbook on the subject. Our bookshelves contain more than a twenty probability texts, many of them directed at elec- trical and computer engineering students. We respect most of them. However, we have yet to ﬁnd one that works well for Rutgers students. We discovered to our surprise that the majority of our students have a hard time learning the subject. Beyond meeting degree requirements, the main motivation of most of our students is to learn how to solve practical problems. For the majority, the mathematical logic of probability theory is, in itself, of minor interest. What the students want most is an intuitive grasp of the basic concepts and lots of practice working on applications. The students told us that the textbooks we assigned, for all their mathematical elegance, didn’t meet their needs. To help them, we distributed copies of our lecture notes, which gradually grew into this book. We also responded to students who ﬁnd that although much of the material appears deceptively simple, it takes a lot of careful thought and practice to vii

viii PREFACE use the mathematics correctly. Even when the formulas are simple, knowing which ones to use is difﬁcult. This is a reversal from some mathematics courses, where the equations are given and the solutions are hard to obtain. What is distinctive about this book? • The entire text adheres to a single model that begins with an experiment consisting of a procedure and observations. • The mathematical logic is apparent to readers. Every fact is identiﬁed clearly as a deﬁnition, an axiom, or a theorem. There is an explanation, in simple English, of the intuition behind every concept when it ﬁrst appears in the text. • The mathematics of discrete random variables are introduced separately from the mathematics of continuous random variables. • Stochastic processes and statistical inference ﬁt comfortably within the unifying model of the text. • An abundance of exercises puts the theory to use. New ideas are augmented with detailed solutions of numerical examples. Each section concludes with a simple quiz to help students gauge their grasp of the new material. The book’s Web site contains complete solutions of all of the quizzes. • Each problem at the end of a chapter is labeled with a reference to a section in the chapter and a degree of difﬁculty ranging from “easy” to “experts only.” • There is considerable support on the World Wide Web for students and instructors, including Matlab ﬁles and problem solutions. How is the book organized? We estimate that the material in this book represents about 150% of a one-semester under- graduate course. We suppose that most instructors will spend about two-thirds of a semester covering the material in the ﬁrst ﬁve chapters. The remainder of a course will be devoted to about half of the material in the ﬁnal seven chapters, with the selection depending on the preferences of the instructor and the needs of the students. Rutgers electrical and computer engineering students take this course in the ﬁrst semester of junior year. The following semester they use much of the material in Principles of Communications. We have also covered the entire book in one semester in an entry-level graduate course that places more emphasis on mathematical derivations and proofs than does the undergraduate course. Although most of the early material in the book is familiar in advance to many graduate students, the course as a whole brings our diverse graduate student population up to a shared level of competence. The ﬁrst ﬁve chapters carry the core material that is common to practically all intro- ductory engineering courses in probability theory. Chapter 1 examines probability models deﬁned on abstract sets. It introduces the set theory notation used throughout the book and states the three axioms of probability and several theorems that follow directly from the ax- ioms. It deﬁnes conditional probability, the Law of Total Probability, Bayes’ theorem, and

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