Schaum's Outline of

1 Schaum's Outline ofTheory and Problems ofProbability, Random Variables, and Random ProcessesHwei P. Hsu, of Electrical Engineering Fairleigh Dickinson UniversityStart 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 from National Taiwan University and and 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 's Outline of Theory and Problems ofPROBABILITY, RANDOM VARIABLES, AND RANDOM PROCESSESC opyright 1997 by The McGraw-Hill Companies, Inc.

2 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 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 BidermanProduction Supervisor: Donald F. SchmidtEditing Supervisor: Maureen WalkerLibrary of Congress Cataloging-in-Publication DataHsu, Hwei P. (Hwei Piao), dateSchaum's Outline of theory and problems of probability , randomvariables, and random processes / Hwei P.

3 Cm. ( Schaum's Outline series)Includes 0-07-030644-31. Probabilities Problems, exercises, etc. 2. Probabilities-Outlines, syllabi, etc. 3. Stochastic processes Problems, exercises, etc. 4. Stochasticprocesses Outlines, syllabi, '076 dc20 96-18245 CIPS tart of Citation[PU]McGraw-Hill Professional[/PU][DP]1997[/DP]End of Citation PrefaceThe purpose of this book is to provide an introduction to principles of probability , random variables, and random processes and their book is designed for students in various disciplines of engineering, science, mathematics, and management.

4 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 wish to thank Dr.

5 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 P. HSUMONTVILLE, NEW JERSEYS tart of Citation[PU]McGraw-Hill Professional[/PU][DP]1997[/DP]End of Citation Contents Chapter 1. Probability1 Introduction1 Sample Space and Events1 Algebra of Sets2 The Notion and Axioms of Probability5 Equally Likely Events7 Conditional Probability7 Total Probability8 Independent Events8 Solved Problems9 Chapter 2.

6 Random Variables38 Introduction38 Random Variables38 Distribution Functions39 Discrete Random Variables and probability Mass Functions41 Continuous Random Variables and probability Density Functions41 Mean and Variance42 Some Special Distributions43 Conditional Distributions48 Solved Problems48 Chapter 3. Multiple Random Variables79 Introduction79 Bivariate Random Variables79 Joint Distribution Functions80 Discrete Random Variables - Joint probability Mass Functions81 Continuous Random Variables - Joint probability Density Functions82 Conditional Distributions83 Covariance and Correlation Coefficient84 Conditional Means and Conditional Variances85 N-Variate Random Variables86 Special Distributions88 Solved Problems89 vviChapter 4.

7 Functions of Random Variables, Expectation, Limit Theorems122 Introduction122 Functions of One Random Variable122 Functions of Two Random Variables123 Functions of n Random Variables124 Expectation125 Moment Generating Functions126 Characteristic Functions127 The Laws of Large Numbers and the Central Limit Theorem128 Solved Problems129 Chapter 5. Random Processes161 Introduction161 Random Processes161 Characterization of Random Processes161 Classification of Random Processes162 Discrete-Parameter Markov Chains165 Poisson Processes169 Wiener Processes172 Solved Problems172 Chapter 6. Analysis and Processing of Random Processes209 Introduction209 Continuity, Differentiation, Integration209 Power Spectral Densities210 White Noise213 Response of Linear Systems to Random Inputs213 Fourier Series and Karhunen-Lo ve Expansions216 Fourier Transform of Random Processes218 Solved Problems219 Chapter 7.

8 Estimation Theory247 Introduction247 Parameter Estimation247 Properties of Point Estimators247 Maximum-Likelihood Estimation248 Bayes' Estimation248 Mean Square Estimation249 Linear Mean Square Estimation249 Solved Problems250 viiChapter 8. Decision Theory264 Introduction264 Hypothesis Testing264 Decision Tests265 Solved Problems268 Chapter 9. Queueing Theory281 Introduction281 Queueing Systems281 Birth-Death Process282 The M/M/1 Queueing System283 The M/M/s Queueing System284 The M/M/1/K Queueing System285 The M/M/s/K Queueing System285 Solved Problems286 Appendix A. Normal Distribution297 Appendix B.

9 Fourier Transform299 Continuous-Time Fourier Transform 299 Discrete-Time Fourier Transform

10 300 Index 303 Chapter 1 probability 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.