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Introduction to Probability.pdf

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    Title Page
    Preface
    Preface to the Second Edition
    CONTENTS
    1. Sample Space and Probability
    1.1 SETS
    1.2 PROBABILISTIC MODELS
    1.3 CONDITIONAL PROBABILITY
    1.4 TOTAL PROBABILITY THEOREM AND BAYES' RULE
    1.5 INDEPENDENCE
    1.6 COUNTING
    SUMMARY AND DISCUSSION
    PROBLEMS
    2. Discrete Random Variables
    2.1 BASIC CONCEPTS
    2.2 PROBABILITY MASS FUNCTIONS
    2.3 FUNCTIONS OF RANDOM VARIABLES
    2.4 EXPECTATION, MEAN, AND VARIANCE
    2.5 JOINT PMFS OF MULTIPLE RANDOM VARIABLES
    2.6 CONDITIONING
    2.7 INDEPENDENCE
    2.8 SUMMARY AND DISCUSSION
    PROBLEMS
    3. General Random Variables
    3.1 CONTINUOUS RANDOM VARIABLES AND PDFS
    3.2 CUMULATIVE DISTRIBUTION FUNCTIONS
    3.3 NORMAL RANDOM VARIABLES
    3.4 JOINT PDFS OF MULTIPLE RANDOM VARIABLES
    3.5 CONDITIONING
    3.6 THE CONTINUOUS BAYES' RULE
    3.7 SUMMARY AND DISCUSSION
    PROBLEMS
    4. Further Topics on Random Variables
    4.1 DERIVED DISTRIBUTIONS
    4.2 COVARIANCE AND CORRELATION
    4.3 CONDITIONAL EXPECTATION AND VARIANCE REVISITED
    4.4 TRANSFORMS
    4.5 SUM OF A RANDOM NUMBER OF INDEPENDENT RANDOMVARIABLES
    4.6 SUMMARY AND DISCUSSION
    PROBLEMS
    5. Limit Theorems
    5.1 MARKOV AND CHEBYSHEV INEQUALITIES
    5.2 THE WEAK LAW OF LARGE NUMBERS
    5.3 CONVERGENCE IN PROBABILITY
    5.4 THE CENTRAL LIMIT THEOREM
    5.5 THE STRONG LAW OF LARGE NUMBERS
    5.6 SUMMARY AND DISCUSSION
    PROBBLEMS
    6. The Bernoulli and Poisson Processes
    6.1 THE BERNOULLI PROCESS
    6.2 THE POISSON PROCESS
    6.3 SUMMARY AND DISCUSSION
    PROBLEMS
    7. Markov Chains
    7.1 DISCRETE-TIME MARKOV CHAINS
    7.2 CLASSIFICATION OF STATES
    7.3 STEADY-STATE BEHAVIOR
    7.4 ABSORPTION PROBABILITIES AND EXPECTED TIMETO ABSORPTION
    7.5 CONTINUOUS-TIME MARKOV CHAINS
    7.6 SUMMARY AND DISCUSSION
    PROBLEMS
    8. Bayesian Statistical Inference
    8.1 BAYESIAN INFERENCE AND THE POSTERIOR DISTRIBUTION
    8.2 POINT ESTIMATION, HYPOTHESIS TESTING, AND THE MAP RULE
    8.3 BAYESIAN LEAST MEAN SQUARES ESTIMATION
    8.4 BAYESIAN LINEAR LEAST MEAN SQUARES ESTIMATION
    8.5 SUMMARY AND DISCUSSION
    PROBLEMS
    9. Classical Statistical Inference
    9.1 CLASSICAL PARAMETER ESTIMATION
    9.2 LINEAR REGRESSION
    9.3 BINARY HYPOTHESIS TESTING
    9.4 SIGNIFICANCE TESTING
    9.5 SUMMARY AND DISCUSSION
    PROBLEMS
    INDEX
    Introduction SECOND EDITION to Probability Dimitri P. Bertsekas and John N. Tsitsiklis Massachusetts Institute of Technology WWW site for book information and orders http://www.athenasc.com ~ Athena Scientific , Belmont, Massachusetts
    Athena Scientific Post Office Box 805 Nashua, NH 03061-0805 U.S.A. Email: info@athenasc.com WWW: http://www.athenasc.com Cover Design: Ann Gallager © 2002, 2008 Dimitri reserved. All rights electronic or mechanical tion storage P. Bertsekas and John N. Tsitsiklis may be reproduced in any form by any No part of this book means (including photocopying, and retrieval) without permission in writing recording, from the publish or informa­ er. Publisher's Cataloging-in-Pub lication Data P., Tsitsiklis, phical John N. to Probability references Bertsekas, Dimitri Introduction Includes bibliogra L Probabilities. 2. Stochastic QA273.B475 2008 519.2 -21 Library of Congress Control ISBN 978-1-886529-23-6 Number: and index 2002092167 Processes. I. Title.
    To the memory of Pantelis Bertsekas and Nikos Tsitsiklis
    Preface Probability is common sense reduced to calculation Laplace course This book is an outgrowth ability of Technology. The course is attended ("Probabi listic of our involvement in teaching an Massachus Analysis'�) at the Systems introductory prob­ etts Institute by a large number of students with diverse back­ spectrum from to the s. They span the entire and from the engineering school graduate and a broad range of interest students, of management. Accordingly, grounds, freshmen to beginning school simplicity has been to develop a manner that combines intuitive understanding in exposition and sophistication the abilit we have tried to strik e a balance between in analytical reasoning. Our key aim y to construct and analyze probabi listic models in and mathematical rigorous analysis has been proofs do not exposition. At the same time, some of precision. In this spirit, some of the more mathematically ely explain or intuitiv way of an otherwise is developed just sketched stand in the this analysis lems, that are included at the end of the corresponding some of the subtler the more attentive ed in the text. so that complex simple level of advanced mathematical issues reader. calculus) are hinted (at the at in footnotes in theoretical prob­ chapter. FUrthermore, addressed to The book covers the fundamentals of probability theory (probabilistic mod­ and continuous random variables, els, discrete multiple limit theorems) , which are typically part of a first course also contains, 4-6 a number of more advanced instructor Chapter of random variables, transforms, a more advanced estimation, to match the goals of a particula 4, we develop in Chapters least squares can choose random variables, and on the subject . It from which an topics, view of conditioning, sums normal distribu- and the bivariate r course. In particular, in v
    vi Preface tion. Furthermore, to Bernoulli, Poisson, and Markov processes in Chapters 5 and 6, we provide . a fairly detailed introduction Our M.LT. course of the material on the normal (Section all seven chapters bivariate semester, with the ex­ 4.7), and on conti in a single covers nuous­ ception time Markov chains on stochastic on foundational (Section 6.5). However, in an alternative processes could be omitted, thereby allowing the material course, additional emphasis material, or coverage of other topics of the instructor's choice. Our most notable omission in coverage all the basic elements and continuous models, and estimation, of parameter is an introduction of Bayesian statistics, in the form of estimation, we hypothesis least squares or non-Bayesian to statistics. develop While we Bayes' rule for discrete do not testing. enter the subjects The problems that supplement the main text are divided in three categories : (a) Theoretical problems: The theoretical problems nent of the text, and ensure (marked by *) constitute that the mathematically reader an important compo oriented Their solutions to solve many solutions. will find here a smooth development are given of them, especially text, but an ambitious reader chapters, in earlier in the without major gaps. may be able before looking at the problems, theoretical of difficulty. These are representative covered the text contains several of the in recitation l sessions and tutoria at mechanism through which many of our students (b) Problems in the text: Besides levels of various that are usually problems, problems M.LT., and are a primary learn the material. solve these problems, enhance the book's www site Our hope is that students elsewhere and then refer to their solutions will attempt to calibrate their understanding of the material. The solutions are posted to and on http://www.athenasc.com/probbo ok.html (c) Supplementary problems: There is a large (and growing) collection of ad­ included in the book, but is made available which is not problems, ditional at the book's homework or exam problems elsewhere these additional available problems will use them for a similar www site. Many of these problems have been assigned as at M.I.T., and we expect that rs of the solutions purpose. accessible, While the statements instructo are made are publicly from the authors only to course instructors. our debt to several people who contributed We would like to acknowledge book. Our writing probability for a popular for several in various ways to the sponsibility had taught zation various used in recitation in AI's classic sessions of the subject that topics textbook, decades. We were thus fortunate had stood the test of time, project began when we assumed re­ class at M.LT. that our colleague Al Drake to start with an organi­ and a rich set of material that a lively presentation of the had been and for homework. We are thus indebted to Al Drake
    Preface vii for providing a very favorable set of initial conditions. We are thankful to the several colleagues who have either taught from the draft of the book with valuable de Veciana. Eugene Feinberg, Ilya Pollak, David Tse, and Terry Wagner. feedback. In particular, we thank Ibrahim at various universities or have read it, and have provided us Abou Faycal, Gustavo Bob Gray, Muriel Medard, Jason Papastavrou, The teaching assistants for the M.LT. class have been very helpful. They to various drafts, they developed problems and solutions pointed out corrections suitable they provided Reaching a robust thousands for the class, and through mechanism interac their direct for calibrating students the level of the material. at MJ. T. at an early stage in their tion with the student body, of bright was a great source of satisfaction for us. We thank them for their valu­ being patient while they were taught from a textbook-in­ studies able feedback and for progress. Last but not least, we are grateful to our families for their support through­ out the course of this long project. P. Bertsekas, dimitrib @mit.edu Dimitri John N. Tsitsikl is, jnt@mit.edu Cambridge, Mass. , May 2002
    v iii Preface Preface to the Second Edition This is a substantial material and the addition by about 25 percent. The main changes of new material. The length revision of the 1st edition, involving are the following: a reorgani zation of the book has increased of old (a) Two new chapters on statistica l inference have been added. one on Bayesian and one on classical main concepts through some key examples. methods. Our philosophy understanding of the main methodologies has been to focus on the and to facilitate (b) Chapters 3 and 4 have been revised, in part to accommodate the new inference material of the tion. Section omitted 4.7 of the from the new edition, 1st edition to streamline chapters and in part ­ the presenta (bivariate normal distribution) has been but is available at the book's website. (c) A number of new examples and end-of-chapter problems have been added. The main objectiv e of the new edition is to provide to give them the option Note that Chapters of including 6-7, and Chapters 8- flexibility to instructors of material, and in particular in their choice an introduction 9 are mutually Furthermore, from Chapter offerings based on this book are: to statistical independent inference. , thus allowing Chapter 4 are needed for Chapters 4 is not needed for Chapters for different paths through the book. Sections 4.2-4.3 8 and 9. Thus, some possible course 5-7, and only (a) Probability and introduction Chapters 1-3, Sections 4.2-4.3, Chapter 5, Chapters to statistical 8-9. inference: (b) Probability and introduction with possibly a few sections to stochastic from Chapter processes: 4. Chapters 1-3 and 5-7, We would like to express our thanks to various colleagues valuable of the material comments on the material tributed ganization Vivek Goyal, Anant Sahai, David Tse, George Verghese, Wyatt have been very helpful in this regard. Finally, we thank Mengdi Wang for her help with for the new chapters. in the new chapters. Ed Coffman, Munther in the 1st edition and problems figures Alan Willsky, and John Dahleh, who have con­ and/or the or­ P. Bertsek Dimitri John N. Tsitsiklis, as, dimitrib@ mit.edu jnt@mit.edu Cambridge, Mass., June 2008
    Contents . . . 1. Sample Space and Probabil ity 1.1. Sets . . . . . . 1 .2. Probabilistic 1 .3. Conditional 1 .4. Total Probability 1 .5. Independence 1 .6. Counting . . . . .. . 1 .7. Summary and Discussion . . . . . . Models. . . . . Probability . . . Theorem and Bayes' Rule Problems . .. . . . . . 2. Discrete Random Variables . . . . . Mass Functions of Random Variables 2.1. Basic Concepts 2.2. Probability 2.3. Functions 2.4. Expectation, 2.5. Joint PMFs of �lultiple 2.6. Conditionin 2 .7. Independence 2.8. Summary and Discussion . .. . . . Mean, and Variance Random Variables g . . . . . . Problems . . . .. .. . 3. General Random Variables 3 . 1 . Continuous Random Variables 3.2. Cumulative ion Functions 3.3. Normal Random Variables 3.4. Joint PDFs of Multiple 3.5. Conditioning . 3.6. The Continuous Bayes' Rule . .. . . . . . . . . Distribut . . . .. . . and PDFs Random Variables . p.1 . p.3 . p.6 p. 18 p. 28 p. 34 p. 44 p.51 p. 53 p. 71 p.72 p. 74 p.80 p. 81 p. 92 p. 97 p. 109 p.1 15 p.1 19 p. 139 p.I40 p.I48 p.I53 p.I58 p.I64 p.I78 ix
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