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Probability, Statistics,
and Random Processes
for Engineers
Fourth Edition
Henry Stark
Illinois Institute of Technology
John W. Woods
Rensselaer Polytechnic Institute
Boston Columbus Indianapolis New York San Francisco Upper Saddle River
Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
To my father P.D. Stark (in memoriam)
From darkness to light (1941–1945)
Henry Stark
To Harriet.
John. W. Woods
Vice President and Editorial Director, ECS:
Marcia J. Horton
Senior Editor: Andrew Gilfillan
Associate Editor: Alice Dworkin
Editorial Assistant: William Opaluch
Marketing Manager: Tim Galligan
Senior Managing Editor: Scott Disanno
Operations Specialist: Lisa McDowell
Production Liason: Irwin Zucker
Art Director: Jayne Conte
Cover Designer: Bruce Kenselaar
Full-Service Project Management/
Composition: Integra Software
Services Pvt. Ltd
Copyright
c
2012, 2002, 1994, 1986 by Pearson Education, Inc., publishing as Prentice Hall. All
rights reserved. Manufactured in the United States of America. This publication is protected by Copyright,
and permission should be obtained from the publisher prior to any prohibited reproduction, storage in
a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying,
recording, or likewise. To obtain permission(s) to use material from this work, please submit a written
request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New
Jersey 07458, or you may fax your request to 201-236-3290.
Many of the designations by manufacturers and seller to distinguish their products are claimed as trade-
marks. Where those designations appear in this book, and the publisher was aware of a trademark claim,
the designations have been printed in initial caps or all caps.
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 these programs
or the documentation 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 these programs.
Library of Congress Cataloging-in-Publication Data on File
10987654321
ISBN 10: 0-13-231123-2
ISBN 13: 978-0-13-231123-6
Contents
Preface xi
1 Introduction to Probability 1
1.1 Introduction: Why Study Probability? 1
1.2 The Different Kinds of Probability 2
Probability as Intuition 2
Probability as the Ratio of Favorable to Total Outcomes (Classical Theory) 3
Probability as a Measure of Frequency of Occurrence 4
Probability Based on an Axiomatic Theory 5
1.3 Misuses, Miscalculations, and Paradoxes in Probability 7
1.4 Sets, Fields, and Events 8
Examples of Sample Spaces 8
1.5 Axiomatic Definition of Probability 15
1.6 Joint, Conditional, and Total Probabilities; Independence 20
Compound Experiments 23
1.7 Bayes’ Theorem and Applications 35
1.8 Combinatorics 38
Occupancy Problems 42
Extensions and Applications 46
1.9 Bernoulli Trials—Binomial and Multinomial Probability Laws 48
Multinomial Probability Law 54
1.10 Asymptotic Behavior of the Binomial Law: The Poisson Law 57
1.11 Normal Approximation to the Binomial Law 63
Summary 65
Problems 66
References 77
iii
iv Contents
2 Random Variables 79
2.1 Introduction 79
2.2 Definition of a Random Variable 80
2.3 Cumulative Distribution Function 83
Properties of F
X
(x)84
Computation of F
X
(x)85
2.4 Probability Density Function (pdf) 88
Four Other Common Density Functions 95
More Advanced Density Functions 97
2.5 Continuous, Discrete, and Mixed Random Variables 100
Some Common Discrete Random Variables 102
2.6 Conditional and Joint Distributions and Densities 107
Properties of Joint CDF F
XY
(x, y) 118
2.7 Failure Rates 137
Summary 141
Problems 141
References 149
Additional Reading 149
3 Functions of Random Variables 151
3.1 Introduction 151
Functions of a Random Variable (FRV): Several Views 154
3.2 Solving Problems of the Type Y = g(X) 155
General Formula of Determining the pdf of Y = g(X) 166
3.3 Solving Problems of the Type Z = g(X, Y ) 171
3.4 Solving Problems of the Type V = g(X, Y ), W = h(X, Y ) 193
Fundamental Problem 193
Obtaining f
VW
Directly from f
XY
196
3.5 Additional Examples 200
Summary 205
Problems 206
References 214
Additional Reading 214
4 Expectation and Moments 215
4.1 Expected Value of a Random Variable 215
On the Validity of Equation 4.1-8 218
4.2 Conditional Expectations 232
Conditional Expectation as a Random Variable 239
4.3 Moments of Random Variables 242
Joint Moments 246
Properties of Uncorrelated Random Variables 248
Jointly Gaussian Random Variables 251
4.4 Chebyshev and Schwarz Inequalities 255
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