Springer Undergraduate Texts in Mathematics and Technology

Series Editors Jonathan M. Borwein Helge Holden Editorial Board Lisa Goldberg Armin Iske Palle E.T. Jorgensen Stephen M. Robinson

Mario Lefebvre Basic Probability Theory with Applications

ISBN 978-0-387-74994-5 e-ISBN 978-0-387-74995-2 DOI 78-0-387-74995-2

Library of Congress Control Number: 2009928845 Mathematics Subject Classification (2000): 60-01 © Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

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Mario Lefebvre Département de mathématiques et de génie industriel École Polytechnique de Montréal, Québec C.P. 6079, succ. Centre-ville Montréal H3C 3A7 Canada mlefebvre@polymtl.ca

10.1007/9Springer Dordrecht Heidelberg London New York

Series EditorsJonathan M. Borwein Helge HoldenFaculty of Computer Science Department of Mathematical SciencesDalhousie University Norwegian University of Science andHalifax, Nova Scotia B3H 1W5 TechnologyCanada Alfred Getz vei 1jborwein@cs.dal.ca NO-7491 Trondheim

Norwayholden@math.ntnu.no

To the memory of my father

I will never believe that God plays dice with the universe.

Albert Einstein

Then they gave lots to them, and the lot fell upon Matthias,and he was counted with the eleven apostles.

Acts 1: 26

Preface

The main intended audience for this book is undergraduate students in pure andapplied sciences, especially those in engineering. Chapters 2 to 4 cover the probabilitytheory they generally need in their training. Although the treatment of the subject issurely sufficient for non-mathematicians, I intentionally avoided getting too much intodetail. For instance, topics such as mixed type random variables and the Dirac deltafunction are only briefly mentioned.

Courses on probability theory are often considered difficult. However, after havingtaught this subject for many years, I have come to the conclusion that one of the biggestproblems that the students face when they try to learn probability theory, particularlynowadays, is their deficiencies in basic differential and integral calculus. Integration byparts, for example, is often already forgotten by the students when they take a courseon probability. For this reason, I have decided to write a chapter reviewing the basicelements of differential calculus. Even though this chapter might not be covered in class,the students can refer to it when needed. In this chapter, an effort was made to give thereaders a good idea of the use in probability theory of the concepts they should alreadyknow.

Chapter 2 presents the main results of what is known as elementary probability,including Bayes’ rule and elements of combinatorial analysis. Although these notionsare not mathematically complicated, it is often a chapter that the students find hardto master. There is no trick other than doing a lot of exercises to become comfortablewith this material.

Chapter 3 is devoted to the more technical subject of random variables. All theimportant models for the applications, such as the binomial and normal distributions,are introduced. In general, the students do better when examined on this subject andfeel that their work is more rewarded than in the case of combinatorial analysis, inparticular.

Random vectors, including the all-important central limit theorem, constitute thesubject of Chapter 4. I have endeavored to present the material as simply as possible.Nevertheless, it is obvious that double integrals cannot be simpler than single integrals.

Applications of Chapters 2 to 4 are presented in Chapters 5 to 7. First, Chapter 5 isdevoted to the important subject of reliability theory, which is used in most engineeringdisciplines, in particular in mechanical engineering. Next, the basic queueing models arestudied in Chapter 6. Queueing theory is needed for many computer science engineeringstudents, as well as for those in industrial engineering. Finally, the last applicationconsidered, in Chapter 7, is the concept of time series. Civil engineers, notably thosespecialized in hydrology, make use of stochastic processes of this type when they wantto model various phenomena and forecast the future values of a given variable, such asthe flow of a river. Time series are also widely used in economy and finance to representthe variations of certain indices.

VII

No matter the level and the background of the students taking a course on probabilitytheory, one thing is always true: as mentioned above, they must try to solve manyexercises before they can feel that they have mastered the theory. To this end, the bookcontains more than 400 exercises, many of which are multiple part questions. At theend of each chapter, the reader will find some solved exercises, whose solutions can befound in Appendix C, followed by a large number of unsolved exercises. Answers to theeven-numbered questions are provided in Appendix D at the end of the book. There arealso many multiple choice questions, whose answers are given in Appendix E.

It is my pleasure to thank all the people I worked with over the years at the EcolePolytechnique de Montreal and who provided me with interesting exercises that wereincluded in this work.

Finally, I wish to express my gratitude to Vaishali Damle, and the entire publishingteam at Springer, for their excellent support throughout this book project.

Mario LefebvreMontreal, July 2008

Preface

Contents

1 Review of differential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Limits and continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Particular integration techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.1 Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Exercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Elementary probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1 Random experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Total probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.6 Combinatorial analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.7 Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.1 Discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.1.2 Continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Important discrete random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.1 Binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2.2 Geometric and negative binomial distributions . . . . . . . . . . . . . . . . 643.2.3 Hypergeometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2.4 Poisson distribution and process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Important continuous random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .List of Tables

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .List of Figures

vi

xiii

xv

X Contents

3.3.1 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.3.2 Gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.3.3 Weibull distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3.4 Beta distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.3.5 Lognormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4 Functions of random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.4.1 Discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.4.2 Continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.5 Characteristics of random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.6 Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4 Random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.1 Discrete random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2 Continuous random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.3 Functions of random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.3.1 Discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.3.2 Continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.3.3 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.4 Covariance and correlation coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5 Limit theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.6 Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5 Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.2 Reliability of systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.2.1 Systems in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1705.2.2 Systems in parallel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1725.2.3 Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.3 Paths and cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.4 Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6 Queueing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.1 Continuous-time Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.2 Queueing systems with a single server . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.2.1 The M/M/1 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.2.2 The M/M/1 model with finite capacity . . . . . . . . . . . . . . . . . . . . . . 207

6.3 Queueing systems with two or more servers . . . . . . . . . . . . . . . . . . . . . . . . . 2126.3.1 The M/M/s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.3.2 The M/M/s/c model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.4 Exercises for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Contents XI

7 Time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2277.2 Particular time series models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.2.1 Autoregressive processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2357.2.2 Moving average processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2447.2.3 Autoregressive moving average processes . . . . . . . . . . . . . . . . . . . . . 249

7.3 Modeling and forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2517.4 Exercises for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

A List of symbols and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

B Statistical tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

C Solutions to “Solved exercises” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

D Answers to even-numbered exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

E Answers to multiple choice questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

List of Tables

3.1 Means and variances of the probability distributions of Sections 3.2 and3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.1 Distribution function of the binomial distribution . . . . . . . . . . . . . . . . . . . . . 276B.1 Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277B.2 Distribution function of the Poisson distribution . . . . . . . . . . . . . . . . . . . . . . 278B.3 Values of the function Φ(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279B.3 Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280B.4 Values of the function Q−1(p) for some values of p . . . . . . . . . . . . . . . . . . . . 280

List of Figures

1.1 Joint density function in Example 1.3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Integration region in Example 1.3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Region A in solved exercise no. 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Venn diagram for Example 2.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Venn diagram for three arbitrary events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Probability of the union of two arbitrary events. . . . . . . . . . . . . . . . . . . . . . . 312.4 Venn diagram for Example 2.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Notion of conditional probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6 System for part (a) of Example 2.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Venn diagram for part (a) of Example 2.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . 342.8 System for part (b) of Example 2.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.9 Example of the law of total probability with n = 3. . . . . . . . . . . . . . . . . . . . 362.10 Example of a tree diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.11 Tree diagram in Example 2.6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.12 Figure for Exercise no. 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.13 Figure for Exercise no. 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.14 Figure for Exercise no. 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.15 Figure for Exercise no. 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.16 Figure for Exercise no. 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Distribution function of the random variable in Example 3.1.1 (ii). . . . . . . 573.2 Density function of the random variable in Example 3.1.3. . . . . . . . . . . . . . 593.3 Distribution function of the random variable in Example 3.1.3. . . . . . . . . . 603.4 Probability functions of binomial random variables. . . . . . . . . . . . . . . . . . . . 623.5 Probability function of a geometric random variable. . . . . . . . . . . . . . . . . . . 643.6 Density function of a normal random variable. . . . . . . . . . . . . . . . . . . . . . . . 713.7 Density functions of various random variables having a gamma

distribution with λ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

XVI List of Figures

3.8 Probability density function of a W(0.8, 0.5) random variable. . . . . . . . . . . 783.9 Probability density functions of a W(0.8, 0.5) (continuous line) and an

Exp(0.32) (broken line) random variables in the interval [5, 20]. . . . . . . . . . 783.10 Probability density function of a uniform random variable on the

interval (a, b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.11 Skewness coefficient of exponential distributions. . . . . . . . . . . . . . . . . . . . . . 913.12 Skewness coefficient of the uniform distribution. . . . . . . . . . . . . . . . . . . . . . . 92

4.1 Joint distribution function in Example 4.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . 1244.2 Density function in Example 4.3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.3 Figure for Exercise no. 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.4 Figure for Exercise no. 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.1 Failure rate function having the shape of a bathtub. . . . . . . . . . . . . . . . . . . 1675.2 A bridge system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.3 A bridge system represented as a parallel system made up of its minimal

path sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.4 A bridge system represented as a series system made up of its minimal

cut sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.5 Figure for Exercise no. 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1875.6 Figure for multiple choice question no. 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.1 State transition diagram for the M/M/1 model. . . . . . . . . . . . . . . . . . . . . . . 2016.2 State transition diagram for the queueing model in Example 6.2.2. . . . . . . 2056.3 State transition diagram for the queueing model in Example 6.2.4. . . . . . . 2126.4 State transition diagram for the M/M/2 model. . . . . . . . . . . . . . . . . . . . . . . 2146.5 Figure for Example 6.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7.1 Joint probability density functions of a random vector having a bivariatenormal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

7.2 Scatter diagram of the data in Example 7.3.1. . . . . . . . . . . . . . . . . . . . . . . . . 255

C.1 Figure for solved exercise no. 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299C.2 Figure for solved exercise no. 27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305