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Schaum's Outline of Theory and Problems of Probability, Random Variables, and Random Processes Hwei P. Hsu, Ph.D. Professor of Electrical Engineering Fairleigh Dickinson University Start of Citation[PU]McGraw-Hill Professional[/PU][DP]1997[/DP]End of Citation HWEI P. HSU is Professor of Electrical Engineering at Fairleigh Dickinson University. He received his B.S. from National Taiwan University and M.S. and Ph.D. from Case Institute of Technology. He has published several books which include Schaum's Outline of Analog and Digital Communications and Schaum's Outline of Signals and Systems. Schaum's Outline of Theory and Problems of PROBABILITY, RANDOM VARIABLES, AND RANDOM PROCESSES Copyright © 1997 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 PRS PRS 9 0 1 0 9 8 7 ISBN 0-07-030644-3 Sponsoring Editor: Arthur Biderman Production Supervisor: Donald F. Schmidt Editing Supervisor: Maureen Walker Library of Congress Cataloging-in-Publication Data Hsu, Hwei P. (Hwei Piao), date Schaum's outline of theory and problems of probability, random variables, and random processes / Hwei P. Hsu. p. cm. \u2014 (Schaum's outline series) Includes index. ISBN 0-07-030644-3 1. Probabilities\u2014Problems, exercises, etc. 2. Probabilities- Outlines, syllabi, etc. 3. Stochastic processes\u2014Problems, exercises, etc. 4. Stochastic processes\u2014Outlines, syllabi, etc. I. Title. QA273.25.H78 1996 519.2'076\u2014dc20 96- 18245 CIP Start of Citation[PU]McGraw-Hill Professional[/PU][DP]1997[/DP]End of Citation Preface The purpose of this book is to provide an introduction to principles of probability, random variables, and random processes and their applications. The book is designed for students in various disciplines of engineering, science, mathematics, and management. It may be used as a textbook and/or as a supplement to all current comparable texts. It should also be useful to those interested in the field for self-study. The book combines the advantages of both the textbook and the so-called review book. It provides the textual explanations of the textbook, and in the direct way characteristic of the review book, it gives hundreds of completely solved problems that use essential theory and techniques. Moreover, the solved problems are an integral part of the text. The background required to study the book is one year of calculus, elementary differential equations, matrix analysis, and some signal and system theory, including Fourier transforms. I wish to thank Dr. Gordon Silverman for his invaluable suggestions and critical review of the manuscript. I also wish to express my appreciation to the editorial staff of the McGraw-Hill Schaum Series for their care, cooperation, and attention devoted to the preparation of the book. Finally, I thank my wife, Daisy, for her patience and encouragement. HWEI P. HSU MONTVILLE, NEW JERSEY Start of Citation[PU]McGraw-Hill Professional[/PU][DP]1997[/DP]End of Citation Contents Chapter 1. Probability 1 1.1 Introduction 1 1.2 Sample Space and Events 1 1.3 Algebra of Sets 2 1.4 The Notion and Axioms of Probability 5 1.5 Equally Likely Events 7 1.6 Conditional Probability 7 1.7 Total Probability 8 1.8 Independent Events 8 Solved Problems 9 Chapter 2. Random Variables 38 2.1 Introduction 38 2.2 Random Variables 38 2.3 Distribution Functions 39 2.4 Discrete Random Variables and Probability Mass Functions 41 2.5 Continuous Random Variables and Probability Density Functions 41 2.6 Mean and Variance 42 2.7 Some Special Distributions 43 2.8 Conditional Distributions 48 Solved Problems 48 Chapter 3. Multiple Random Variables 79 3.1 Introduction 79 3.2 Bivariate Random Variables 79 3.3 Joint Distribution Functions 80 3.4 Discrete Random Variables - Joint Probability Mass Functions 81 3.5 Continuous Random Variables - Joint Probability Density Functions 82 3.6 Conditional Distributions 83 3.7 Covariance and Correlation Coefficient 84 3.8 Conditional Means and Conditional Variances 85 3.9 N-Variate Random Variables 86 3.10 Special Distributions 88 Solved Problems 89 v vi Chapter 4. Functions of Random Variables, Expectation, Limit Theorems 122 4.1 Introduction 122 4.2 Functions of One Random Variable 122 4.3 Functions of Two Random Variables 123 4.4 Functions of n Random Variables 124 4.5 Expectation 125 4.6 Moment Generating Functions 126 4.7 Characteristic Functions 127 4.8 The Laws of Large Numbers and the Central Limit Theorem 128 Solved Problems 129 Chapter 5. Random Processes 161 5.1 Introduction 161 5.2 Random Processes 161 5.3 Characterization of Random Processes 161 5.4 Classification of Random Processes 162 5.5 Discrete-Parameter Markov Chains 165 5.6 Poisson Processes 169 5.7 Wiener Processes 172 Solved Problems 172 Chapter 6. Analysis and Processing of Random Processes 209 6.1 Introduction 209 6.2 Continuity, Differentiation, Integration 209 6.3 Power Spectral Densities 210 6.4 White Noise 213 6.5 Response of Linear Systems to Random Inputs 213 6.6 Fourier Series and Karhunen-Loéve Expansions 216 6.7 Fourier Transform of Random Processes 218 Solved Problems 219 Chapter 7. Estimation Theory 247 7.1 Introduction 247 7.2 Parameter Estimation 247 7.3 Properties of Point Estimators 247 7.4 Maximum-Likelihood Estimation 248 7.5 Bayes' Estimation 248 7.6 Mean Square Estimation 249 7.7 Linear Mean Square Estimation 249 Solved Problems 250 vii Chapter 8. Decision Theory 264 8.1 Introduction 264 8.2 Hypothesis Testing 264 8.3 Decision Tests 265 Solved Problems 268 Chapter 9. Queueing Theory 281 9.1 Introduction 281 9.2 Queueing Systems 281 9.3 Birth-Death Process 282 9.4 The M/M/1 Queueing System 283 9.5 The M/M/s Queueing System 284 9.6 The M/M/1/K Queueing System 285 9.7 The M/M/s/K Queueing System 285 Solved Problems 286 Appendix A. Normal Distribution 297 Appendix B. Fourier Transform 299 B.1 Continuous-Time Fourier Transform 299 B.2 Discrete-Time Fourier Transform 300 Index 303 Chapter 1 Probability 1.1 INTRODUCTION The study of probability stems from the analysis of certain games of chance, and it has found applications in most branches of science and engineering. In this chapter the basic concepts of prob- ability theory are presented. 1.2 SAMPLE SPACE AND EVENTS A. Random Experiments: In the study of probability, any process of observation is referred to as an experiment. The results of an observation are called the outcomes of the experiment. An experiment is called a random experi- ment if its outcome cannot be predicted. Typical examples of a random experiment are the roll of a die, the toss of a coin, drawing a card from a deck, or selecting a message signal for transmission from several messages. B. Sample Space: The set of all possible outcomes of a random experiment is called the sample space (or universal set), and it is denoted by S. An element in S is called