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Elementary Probability
2ndEdition
Now available in a fully revised and updated new edition, this well-established text-
bookprovidesastraightforwardintroductiontothetheoryofprobability.Thepresen-
tation is entertaining without any sacrifice of rigour; important notions are covered
withtheclaritythatthesubjectdemands.
Topicscoveredincludeconditionalprobability,independence,discreteandcontin-
uousrandomvariables,basiccombinatorics,generatingfunctionsandlimittheorems,
andanintroductiontoMarkovchains.Thiseditionincludesanelementaryapproach
tomartingalesandthetheoryofBrownianmotion,whichsupplythecornerstonesfor
many topics in modern financial mathematics such as option and derivative pricing.
Thetextisaccessibletoundergraduatestudents,andprovidesnumerousworkedex-
amplesandexercisestohelpbuildtheimportantskillsnecessaryforproblemsolving.
‘[T]heauthorsucceedsincombiningtheutmostsuccinctnesswithclarityandgenuineread-
ability....Thistextbookcanberecommendedunreservedly.’
InternationaleMathematischeNachrichten
‘[T]hisbookisasuperbresourceoftheoryandapplication,whichshouldbeoneverylecturer’s
shelves,andthoseofmanystudents.Youmayneverneedtobuyanotherbookonprobability.’
KeithHirst,TheMathematicalGazette
‘Excellent! A vast number of well-chosen worked examples and exercises guide the reader
throughthebasictheoryofprobabilityattheelementarylevel...anexcellenttextwhichIam
surewillgivealotofpleasuretostudentsandteachersalike.’
InternationalStatisticsInstitute
‘[W]ouldmakeafineadditiontoanundergraduatelibrary.Astudentwithasolidbackground
incalculus,linearalgebra,andsettheorywillfindmanyusefultoolsofelementaryprobability
here.’
PhilGilbert,TheMathematicsTeacher
‘Stirzakerdoesanexcellentjobofdevelopingproblem-solvingskillsinanintroductoryproba-
bilitytext.Numerousexamplesandpracticeexercisesareprovidedthatonlyservetoenhance
astudent’sproblem-solvingabilities....Highlyrecommended.’
D.J.Gougeon,Choice
‘The book would make an excellent text for the properly prepared class, a solid instructor’s
referenceforbothprobabilityapplicationsandproblems,aswellasafineworkforpurposes
ofself-study.’
J.PhilipSmith,SchoolScienceandMathematics
Elementary Probability
2nd Edition
by
D A V I D S T I R Z A K E R
MathematicalInstituteandSt.John’sCollege,
UniversityofOxford
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521833448
© David Stirzaker 2003
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2003
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Cambridge University Press has no responsibility for the persistence or accuracy of
s for external or third-party internet websites referred to in this book, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
PrefacetotheSecondEdition pagexi
0 Introduction 1
0.1 Chance 1
0.2 Models 3
0.3 Symmetry 5
0.4 TheLongRun 7
0.5 Pay-Offs 8
0.6 Introspection 9
0.7 FAQs 10
0.8 History 14
Appendix:ReviewofElementaryMathematicalPrerequisites 15
1 Probability 24
1.1 NotationandExperiments 24
1.2 Events 26
1.3 TheAdditionRulesforProbability 32
1.4 PropertiesofProbability 34
1.5 SequencesofEvents 36
1.6 Remarks 37
1.7 ReviewandChecklistforChapter1 38
Workedexamplesandexercises 40
1.8 Example:Dice 40
1.9 Example:Urn 41
1.10 Example:CupsandSaucers 42
1.11 Example:Sixes 43
1.12 Example:FamilyPlanning 44
1.13 Example:Craps 45
1.14 Example:Murphy’sLaw 46
Problems 47
2 ConditionalProbabilityandIndependence 51
2.1 ConditionalProbability 51
2.2 Independence 57
2.3 RecurrenceandDifferenceEquations 60
2.4 Remarks 62
v
vi Contents
2.5 ReviewandChecklistforChapter2 64
Workedexamplesandexercises 65
2.6 Example:SuddenDeath 65
2.7 Example:Polya’sUrn 66
2.8 Example:Complacency 67
2.9 Example:Dogfight 68
2.10 Example:Smears 69
2.11 Example:Gambler’sRuin 70
2.12 Example:AccidentsandInsurance 72
2.13 Example:Protocols 73
2.14 Example:Eddington’sControversy 75
Problems 76
3 Counting 83
3.1 FirstPrinciples 83
3.2 Permutations:OrderedSelection 84
3.3 Combinations:UnorderedSelection 86
3.4 Inclusion–Exclusion 87
3.5 RecurrenceRelations 88
3.6 GeneratingFunctions 90
3.7 Techniques 93
3.8 ReviewandChecklistforChapter3 95
Workedexamplesandexercises 97
3.9 Example:RailwayTrains 97
3.10 Example:GenoeseLottery 98
3.11 Example:RingingBirds 99
3.12 Example:Lottery 101
3.13 Example:TheMe´nagesProblem 101
3.14 Example:Identity 102
3.15 Example:Runs 103
3.16 Example:Fish 105
3.17 Example:Colouring 106
3.18 Example:Matching(Rencontres) 107
Problems 108
4 RandomVariables:DistributionandExpectation 114
4.1 RandomVariables 114
4.2 Distributions 115
4.3 Expectation 120
4.4 ConditionalDistributions 127
4.5 SequencesofDistributions 130
4.6 Inequalities 131
4.7 ReviewandChecklistforChapter4 134
Workedexamplesandexercises 137
4.8 Example:RoyalOakLottery 137
4.9 Example:Misprints 138
4.10 Example:DogBites:PoissonDistribution 139
Contents vii
4.11 Example:Guesswork 141
4.12 Example:GamblersRuinedAgain 142
4.13 Example:Postmen 143
4.14 Example:AcmeGadgets 144
4.15 Example:RouletteandtheMartingale 145
4.16 Example:Searching 146
4.17 Example:Duelling 147
4.18 BinomialDistribution:TheLongRun 149
4.19 Example:UncertaintyandEntropy 150
Problems 151
5 RandomVectors:IndependenceandDependence 158
5.1 JointDistributions 158
5.2 Independence 162
5.3 Expectation 165
5.4 SumsandProductsofRandomVariables:Inequalities 172
5.5 Dependence:ConditionalExpectation 177
5.6 SimpleRandomWalk 183
5.7 Martingales 190
5.8 TheLawofAverages 196
5.9 Convergence 199
5.10 ReviewandChecklistforChapter5 203
Workedexamplesandexercises 206
5.11 Example:Golf 206
5.12 Example:JointLives 208
5.13 Example:Tournament 209
5.14 Example:Congregations 210
5.15 Example:Propagation 211
5.16 Example:InformationandEntropy 212
5.17 Example:Cooperation 214
5.18 Example:StrangeButTrue 215
5.19 Example:Capture–Recapture 216
5.20 Example:VisitsofaRandomWalk 218
5.21 Example:Ordering 219
5.22 Example:MoreMartingales 220
5.23 Example:SimpleRandomWalkMartingales 221
5.24 Example:YouCan’tBeattheOdds 222
5.25 Example:MatchingMartingales 223
5.26 Example:Three-HandedGambler’sRuin 224
Problems 226
6 GeneratingFunctionsandTheirApplications 232
6.1 Introduction 232
6.2 MomentsandtheProbabilityGeneratingFunction 236
6.3 SumsofIndependentRandomVariables 239
6.4 MomentGeneratingFunctions 245
6.5 JointGeneratingFunctions 247
viii Contents
6.6 Sequences 251
6.7 Regeneration 254
6.8 RandomWalks 259
6.9 ReviewandChecklistforChapter6 263
Appendix:Calculus 265
Workedexamplesandexercises 268
6.10 Example:Gambler’sRuinandFirstPassages 268
6.11 Example:“Fair”PairsofDice 269
6.12 Example:BranchingProcess 271
6.13 Example:GeometricBranching 272
6.14 Example:Waring’sTheorem:OccupancyProblems 274
6.15 Example:BernoulliPatternsandRuns 275
6.16 Example:WaitingforUnusualLightBulbs 277
6.17 Example:MartingalesforBranching 278
6.18 Example:Wald’sIdentity 279
6.19 Example:TotalPopulationinBranching 280
Problems 281
7 ContinuousRandomVariables 287
7.1 DensityandDistribution 287
7.2 FunctionsofRandomVariables 297
7.3 SimulationofRandomVariables 301
7.4 Expectation 302
7.5 MomentGeneratingFunctions 306
7.6 ConditionalDistributions 310
7.7 AgeingandSurvival 312
7.8 StochasticOrdering 314
7.9 RandomPoints 315
7.10 ReviewandChecklistforChapter7 318
Workedexamplesandexercises 321
7.11 Example:UsingaUniformRandomVariable 321
7.12 Example:NormalDistribution 323
7.13 Example:Bertrand’sParadox 324
7.14 Example:StockControl 326
7.15 Example:ObtainingYourVisa 327
7.16 Example:Pirates 329
7.17 Example:FailureRates 330
7.18 Example:Triangles 330
7.19 Example:Stirling’sFormula 332
Problems 334
8 JointlyContinuousRandomVariables 337
8.1 JointDensityandDistribution 337
8.2 ChangeofVariables 342
8.3 Independence 344
8.4 Sums,Products,andQuotients 348
8.5 Expectation 351
8.6 ConditionalDensityandExpectation 355
