intuitive probability and random processes using …978-0-387-24158-6/1 · intuitive probability...

INTUITIVE PROBABILITYANDRANDOM PROCESSESUSING MATLAB®

INTUITIVE PROBABILITYANDRANDOM PROCESSESUSING MATLAB®

STEVEN M. KAYUniversity ofRhode Island

~ Springer

Author: Steven M. Kay University of Rhode Island Dept. of Electrical & Computer Engineering Kingston, RI 02881

2006 Steven M. Kay (4th corrected version of the

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. Printed on acid free paper 9 8 7 6 5 springer.com

5th printing (2012))

ISBN 978-0-387-24157-9 e-ISBN 978-0-387-24158-6 Library of Congress Control Number: 2005051721

To my wifeCindy,

whose love and supportare without measure

and to my daughtersLisa and Ashley,

who are a source of joy

NOTE TO INSTRUCTORS

As an aid to instructors interested in using this book for a course, the solutions tothe exercises are available in electronic form. They may be obtained by contactingthe author at [email protected].

Preface

The subject of probability and random processes is an important one for a variety ofdisciplines. Yet, in the author's experience, a first exposure to this subject can causedifficulty in assimilating the material and even more so in applying it to practicalproblems of interest. The goal of this textbook is to lessen this difficulty. To doso we have chosen to present the material with an emphasis on conceptualization.As defined by Webster, a concept is "an abstract or generic idea generalized fromparticular instances." This embodies the notion that the "idea" is something wehave formulated based on our past experience. This is in contrast to a theorem,which according to Webster is "an idea accepted or proposed as a demonstrabletruth". A theorem then is the result of many other persons' past experiences, whichmayor may not coincide with our own. In presenting the material we prefer tofirst present "part icular instances" or examples and then generalize using a definition/theorem. Many textbooks use the opposite sequence, which undeniably iscleaner and more compact, but omits the motivating examples that initially ledto the definition/theorem. Furthermore, in using the definition/theorem-first approach, for the sake of mathematical correctness multiple concepts must be presentedat once. This is in opposition to human learning for which "under most conditions,the greater the number of attributes to be bounded into a single concept , the moredifficult the learning becomes" 1 . The philosophical approach of specific examplesfollowed by generalizations is embodied in this textbook. It is hoped that it willprovide an alternative to the more traditional approach for exploring the subject ofprobability and random processes.

To provide motivating examples we have chosen to use MATLAB2 , which is avery versatile scientific programming language. Our own engineering students at theUniversity of Rhode Island are exposed to MATLAB as freshmen and continue to useit throughout their curriculum. Graduate students who have not been previouslyintroduced to MATLAB easily master its use. The pedagogical utility of usingMATLAB is that:

1. Specific computer generated examples can be constructed to provide motivationfor the more general concepts to follow.

lEli Sal t z, Th e Cogniti ve Basis of Human Learning, Dorsey Press, Homewood, IL, 1971.2Registered trademark of TheMathWorks, Inc.

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2. Inclusion of computer code within the text allows the reader to interpret themathematical equations more easily by seeing them in an alternative form.

3. Homework problems based on computer simulations can be assigned to illustrateand reinforce important concepts.

4. Computer experimentation by the reader is easily accomplished.

5. Typical results of probabilistic-based algorithms can be illustrated.

6. Real-world problems can be described and "solved" by implementing the solutionin code.

Many MATLAB programs and code segments have been included in the book. Infact, most of the figures were generated using MATLAB . The programs and codesegments listed within the book are available in the file pr'obbook.matLab.code . tex,which can be found at http://www.ele.uri.edu/faculty/kay/New%20web/Books.htm.The use of MATLAB, along with a brief description of its syntax, is introduced earlyin the book in Chapter 2. It is then immediately applied to simulate outcomes ofrandom variables and to estimate various quantities such as means, variances, probability mass functions, etc. even though these concepts have not as yet been formallyintroduced. This chapter sequencing is purposeful and is meant to expose the readerto some of the main concepts that will follow in more detail later. In addition,the reader will then immediately be able to simulate random phenomena to learnthrough doing, in accordance with our philosophy. In summary, we believe thatthe incorporation of MATLAB into the study of probability and random processesprovides a "hands-on" approach to the subject and promotes better understanding.

Other pedagogical features of this textbook are the discussion of discrete randomvariables first to allow easier assimilation of the concepts followed by a parallel discussion for continuous random variables. Although this entails some redundancy, wehave found less confusion on the part of the student using this approach. In a similarvein, we first discuss scalar random variables, then bivariate (or two-dimensional)random variables, and finally N-dimensional random variables, reserving separatechapters for each. All chapters, except for the introductory chapter, begin with asummary of the important concepts and point to the main formulas of the chapter, and end with a real-world example. The latter illustrates the utility of thematerial just studied and provides a powerful motivation for further study. It alsowill, hopefully, answer the ubiquitous question "Why do we have to study this?" .We have tried to include real-world examples from many disciplines to indicate thewide applicability of the material studied. There are numerous problems in eachchapter to enhance understanding with some answers listed in Appendix E. Theproblems consist of four types. There are "formula" problems, which are simple applications of the important formulas of the chapter; "word" problems, which requirea problem-solving capability; and "theoretical" problems, which are more abstract

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and mathematically demanding; and finally, there are "computer" problems, whichare either computer simulations or involve the application of computers to facilitateanalytical solutions. A complete solutions manual for all the problems is availableto instructors from the author upon request. Finally, we have provided warnings onhow to avoid common errors as well as in-line explanations of equations within thederivations for clarification.

The book was written mainly to be used as a first-year graduate level coursein probability and random processes. As such, we assume that the student hashad some exposure to basic probability and therefore Chapters 3-11 can serve asa review and a summary of the notation. We then will cover Chapters 12-15 onprobability and selected chapters from Chapters 16-22 on random processes. Thisbook can also be used as a self-contained introduction to probability at the seniorundergraduate or graduate level. It is then suggested that Chapters 1-7, 10, 11 becovered. Finally, this book is suitable for self-study and so should be useful to thepractitioner as well as the student. The necessary background that has been assumedis a knowledge of calculus (a review is included in Appendix B) ; some linear/matrixalgebra (a review is provided in Appendix C); and linear systems, which is necessaryonly for Chapters 18-20 (although Appendix D has been provided to summarize andillustrate the important concepts).

The author would like to acknowledge the contributions of the many people whoover the years have provided stimulating discussions of teaching and research problems and opportunities to apply the results of that research. Thanks are due to mycolleagues L. Jackson, R. Kumaresan, L. Pakula, and P. Swaszek of the Universityof Rhode Island. A debt of gratitude is owed to all my current and former graduatestudents. They have contributed to the final manuscript through many hours ofpedagogical and research discussions as well as by their specific comments and questions. In particular, Lin Huang and Cuichun Xu proofread the entire manuscript andhelped with the problem solutions, while Russ Costa provided feedback. Lin Huangalso aided with the intricacies of LaTex while Lisa Kay and Jason Berry helped withthe artwork and to demystify the workings of Adobe Illustrator 10.3 The authoris indebted to the many agencies and program managers who have sponsored hisresearch, including the Naval Undersea Warfare Center, the Naval Air Warfare Center, the Air Force Office of Scientific Research, and the Office of Naval Research.As always, the author welcomes comments and corrections, which can be sent [email protected].

Steven M. KayUniversity of Rhode IslandKingston, RI 02881

3Registered trademark of Adobe Systems Inc.

Contents

Preface vii

1 Introduction 11.1 What Is Probability? . . . . . . 11.2 Types of Probability Problems 31.3 Probabilistic Modeling . . . . . 41.4 Analysis versus Computer Simulation 71.5 Some Notes to the Reader 8

References . 9Problems 10

2 Computer Simulation 132.1 Introduction . . .. .... . . .. 132.2 Summary . . . . . . . . . . . . . 132.3 Why Use Computer Simulation? 142.4 Computer Simulation of Random Phenomena 172.5 Determining Characteristics of Random Var iables . 182.6 Real-World Example - Digit al Communications . 24

References . . . . . . . . . . . . . 26Problems 26

2A Brief Introducti on to MATLAB . 31

3 Basic Probability 373.1 Introduction. . 373.2 Summary . . . 373.3 Review of Set Theory 383.4 Assigning and Determining Probabilities. 433.5 Properties of the Probabili ty Function . . 483.6 Probabilities for Continuous Sample Spaces 523.7 Prob abiliti es for Finite Sample Spaces - Equally Likely Ou tcomes 543.8 Combinatorics 553.9 Binomial Probability Law . . . . . . . . . . . . . . . . . . . . . .. 62

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3.10 Real-World Example - Quality ControlReferences .Problems .

4 Conditional Probability4.1 Introduction. . . . . . . . . .. ... .. ...4.2 Summary . . . . . . . . . . . . . . . . . . . .4.3 Joint Events and the Conditional Probability4.4 St atistically Independent Event s4.5 Bayes' Theorem . . . . . . . . . . . . . . . .4.6 Multiple Exp eriment s .4.7 Real-World Example - Cluster Recognition

References .Problems .

5 Discrete Random Variables5.1 Introduction .5.2 Summary . . . . . . . . . . . . . . . . .5.3 Definition of Discrete Random Variable5.4 Probability of Discrete Random Variables5.5 Important Probability Mass Functions . .5.6 Approximation of Binomial PMF by Poisson P MF5.7 Transformation of Discrete Random Variables .5.8 Cumulati ve Distributi on Funct ion .5.9 Computer Simul ation .5.10 Real-World Example - Servicing Customers


6 Expected Values for Discrete Random Variables6.1 Introduction .6.2 Summary . . . . . . . . . . . . . . . . . . . . . . .6.3 Determining Averages from the PMF .6.4 Expected Values of Some Important Random Vari ables6.5 Expected Value for a Function of a Random Vari able.6.6 Variance and Moments of a Random Variable6.7 Characteristic Functions .6.8 Estimating Means and Varian ces .6.9 Real- World Example - Dat a Compression

References . . . . . . . . . . . .Problems .

6A Derivation of E [g(X )] Formula .6B MAT LAB Code Used to Estimate Mean and Variance

CONTENTS

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105105105106108111113115117122124128128

133133133134137140143147153155157158163165

CONTENTS

7 Multiple Discrete Random Variables7.1 Introduction .7.2 Summary .7.3 Jointly Distributed Random Variables7.4 Marginal PMFs and CDFs .7.5 Independence of Multiple Random Variables.7.6 Transformations of Multiple Random Variables7.7 Expected Values .7.8 Joint Moments .7.9 Prediction of a Random Variable Outcome .7.10 Joint Characteristic Functions .7.11 Computer Simulation of Random Vectors .7.12 Real-World Example - Assessing Health Risks .

References . . . . . . . . . . . . . . . . . . . .Problems .

7A Derivation of the Cauchy-Schwarz Inequality

8 Conditional Probabili t y Mass Functions8.1 Introduction .8.2 Summary . . . . . . . . . . . . . . . . .8.3 Conditional Probability Mass Function .8.4 Joint, Conditional, and Marginal P MFs8.5 Simplifying Probability Calculations using Conditioning8.6 Mean of the Conditional PMF . . . . . . . . . . . .8.7 Computer Simulation Based on Conditioning . . .8.8 Real-World Example - Mod eling Human Learning


9 Discrete N -D im ension a l Random Variables9.1 Introduct ion .9.2 Summary . . . . . . . . . . . . . . . . . . .9.3 Random Vectors and Probability Mass Functions9.4 Transformations .9.5 Expected Values .9.6 Joint Moments and the Characteristic Function9.7 Conditional Probability Mass Functions .9.8 Computer Simulation of Random Vectors9.9 Real-World Example - Image Coding .


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XIV CONTENTS

10 Continuous Random Variables 28510.1 Introduction. . . . . . . . . . . . . . . . . . . 28510.2 Summary . . . . . . . . . . . . . . . . . . . . 28610.3 Definition of a Continuous Random Vari able 28710.4 The PDF and Its Properties . . . . 29310.5 Important PDFs . . . . . . . . . . 29510.6 Cumulative Distribution Functions 30310.7 Transformations . ... . 31110.8 Mixed Random Vari ables . . . . . 31710.9 Computer Simulation. . . . . . . . 32410.10Real-World Example - Setting Clipping Levels 328

References. . . . . . . . . . . . . . . . . . . . . 331Problems ... . . . . . . . . . . . . . . . . . . 331

lOA Derivation of PDF of a Transformed Continuous Random Variable 339lOB MATLAB Subprograms to Compute Q and Inverse Q Functions . 341

11 Expected Values for Continuous Random Variables 34311.1 Introduction. . . . . . . . . . . . 34311.2 Summary . . . . . . . . . . . . . . . . 34311.3 Determining the Exp ected Value . . . 34411.4 Expected Values for Imp ortant PDFs . 34911.5 Expected Valu e for a Function of a Random Vari able. 35111.6 Variance and Moments . . . . . . . . . . . . . . . . . 35511.7 Characteristic Functions . . . . . . . . . . . . . . . . 35911.8 Probability, Moments, and the Chebyshev Inequali ty 36111.9 Estimating the Mean and Variance . . . . . . . . 36311.10Real-World Example - Critical Software Testing 364

References . . . . . . . . . . . . . . . . . . . . . . 367Problems 367

11A Partial Proof of Expected Value of Function of Continuous RandomVariable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

12 Multiple Continuous Random Variables 37712.1 Introduction . . . . . . . . . . . . . . . 37712.2 Summary . . . . . . . . . . . . . . . . 37812.3 Jointly Distributed Random Variables 37912.4 Marginal PDFs and the Joint CDF . . 38712.5 Independence of Multiple Random Vari ables. 39212.6 Transformations 39412.7 Expected Values . . . . . . . . . . . . . . 40412.8 Joint Moments . . . . . . . . . . . . . . . 41212.9 Prediction of Random Variable Outcome. 41212.lOJoint Characterist ic Functions . . . . . . . 414

CONTENTS xv

12.11Computer Simulation. . . . . . . . . . . . . . . . . . . 41512.12Real-World Example - Optical Character Recognition 419


13 Conditional Probability Density Functions 43313.1 Int roduct ion . . . 43313.2 Summary . . . . . . . . . . . . . . . . . 43313.3 Conditional P DF . . . . . . . . . . . . . 43413.4 Joint , Conditional, and Marginal PDFs 44013.5 Simplifying Probability Calculations Using Conditioning . 44413.6 Mean of Conditional P DF . . . . . . . . . . . . . . . . . . 44613.7 Computer Simulation of Jointly Continuous Random Variables 44713.8 Real-World Example - Retirement Planning . 449


14 Continuous N-D im ensional Random Variables 45714.1 Introduction . . . . . . . . . 45714.2 Summary . . . . . . . . . . 45714.3 Random Vectors and P DFs 45814.4 Transformations 46314.5 Expected Values . . 46514.6 Joint Moments and the Characteristic Function 46714.7 Conditional P DFs 47114.8 Prediction of a Random Variable Outcome . . . 47114.9 Computer Simulation of Gaussian Random Vectors 47514.10Real-World Example - Signal Detection 476


15 Probability and Moment Approximations U sing Limit Theorems 48515.1 Introduction . . . . . . . . . . . . . . . . . . 48515.2 Summary . . . . . . . . . . . . . . . . . . . 48615.3 Convergence and Approximation of a Sum. 48615.4 Law of Large Numbers . . . . . . . . . . 48715.5 Central Limit Theorem 49215.6 Real-World Example - Opinion Polling . 503

References . . . . . . . . . . . . . . . . . 506Problems 507

15A MATLAB P rogram to Compute Rep eated Convolution of PDFs 51115B Proof of Central Limit Theorem . . . . . . . . . . . . . . . . . . 513

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16 Basic Random Processes16.1 Int roduct ion .16.2 Summary . . . . . . . .16.3 What Is a Random P rocess? .16.4 Types of Random Processes .16.5 The Important Property of Stationarity16.6 Some More Examples .16.7 Joint Moments .16.8 Real-World Example - Statistical Data Analysis


17 Wide Sense Stationary Random Processes17.1 Introduct ion .17.2 Summary .17.3 Definit ion of WSS Random P rocess.17.4 Autocorrelation Sequence .. .. .17.5 Ergodicity and Temporal Averages17.6 The Power Spectral Density ... .17.7 Estimation of the ACS and P SD .17.8 Continuous-Time WSS Random Processes17.9 Real-World Example - Random Vibration Test ing


CONTENTS

515515516517520523528533538542542

547547548549552562567576580586589590

18 Linear Systems and W ide Sense Stationar y Random Processes 59718.1 Introduction . . . . . . . . . . . . . . . . . . . 59718.2 Summary . . . . . . . . . . . . . . . . . . . . 59818.3 Random Process at Output of Linear System 59818.4 Interpretation of the PS D . . . . . . . . . . 60718.5 Wiener Fil tering . . . . . . . . . . . . . . . 60918.6 Continuous-Ti me Definitions and Formulas 62318.7 Real-World Example - Speech Synthesis 626

References . . . . . . . . . . . . . . . . 630Problems 631

18A Solution for Infinite Length Predictor . 637

19 M u lt ip le Wide Sense Stationary Random Processes 64119.1 Introduction . . . . . . . . . . . . . . . . . . 64119.2 Summary . . . . ' .' . . . . . . . . . . . . . 64219.3 Jointly Distributed WSS Random Processes 64219.4 The Cross-Power Spectral Density . . . . . 64719.5 Transformations of Multiple Random Processes 652

CONTENTS

19.6 Continuous-Time Definitions and Formulas19.7 Cross-Correlation Sequence Estimation. . .19.8 Real-World Example - Br ain Physiology Research


20 Gaussian Random Processes20.1 Introduction .20.2 Summary . . . . . . . . . . . . . . . . . . .20.3 Definition of the Gaussian Random Process20.4 Linear Transformations .20.5 Nonlinear Transformations .20.6 Continuous-Time Definitions and Formulas20.7 Special Continuous-Time Gaussian Random Processes20.8 Computer Simulation . . . . . . . . . . . . . . . . . .20.9 Real-World Example - Estimating Fish Populations

References . . . . . . . . . . . . .Problems .

20A MATLAB Listing for Figure 20.2

21 Poisson Random Processes21.1 Introduction .21.2 Summary . . . . . . . . . . . . . . . . . . . . . .21.3 Derivation of Poisson Count ing Random Process21.4 Interar rival Times .21.5 Arr ival Times . . . . . . . . . . . . .21.6 Compound Poisson Random Process21.7 Computer Simulation. . . . . . . . .21.8 Real-World Example - Automobile Traffic Signa l Planning.

References . . . . . . . . . . . . .Problems .

21A Joint PDF for Interarrival Times

22 Markov Chains22.1 Introduction .22.2 Summary . .22.3 Definitions . .22.4 Computation of St at e P robabilities22.5 Ergodic Markov Chains .22.6 Further Stead y-State Characteristics22.7 K-State Markov Chains .22.8 Computer Simulation . . . . . . . . .22.9 Real-World Example - St range Markov Chain Dynamics.

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References . . . . . . . . . . . .Problems .

22A Solving for the Stationary PMF

A Glossary of Symbols and Abbrevations

B Assorted Math Facts and FormulasB.1 Proof by InductionB.2 TrigonometryB.3 Limits .Bo4 Sums ..B.5 Calculus

C Linear and Matrix AlgebraC.1 Definitions .C.2 Special Matrices .C.3 Matrix Manipulation and Formulas .Co4 Some Properties of PD (PSD) MatricesC.5 Eigendecomposition of Matrices . . . . .

CONTENTS

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789789791792793793

D Summary of Signals, Linear Transforms, and Linear Systems 795D.1 Discrete-Time Signals . . . . . 795D.2 Linear Transforms . . . . . . . 796D.3 Discrete-Time Linear Systems. 800DA Continuous-Time Signals. . . . 804D.5 Linear Transforms . . . . . . . 805D.6 Continuous-Time Linear Systems 807

E Answers to Selected Problems 809

Index 823

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