Grinstead, Snell - Introduction to Probability.pdf

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IntroductiontoProbability
CharlesM.Grinstead
SwarthmoreCollege
J.LaurieSnell
DartmouthCollege
Toourwives
andinmemoryof
ReeseT.Prosser
Contents
1DiscreteProbabilityDistributions 1
1.1SimulationofDiscreteProbabilities................... 1
1.2DiscreteProbabilityDistributions.................... 18
2ContinuousProbabilityDensities 41
2.1SimulationofContinuousProbabilities................. 41
2.2ContinuousDensityFunctions...................... 55
3Combinatorics 75
3.1Permutations............................... 75
3.2Combinations............................... 92
3.3CardShuing...............................120
4ConditionalProbability 133
4.1DiscreteConditionalProbability....................133
4.2ContinuousConditionalProbability...................162
4.3Paradoxes.................................175
5DistributionsandDensities 183
5.1ImportantDistributions.........................183
5.2ImportantDensities ...........................205
6ExpectedValueandVariance 225
6.1ExpectedValue..............................225
6.2VarianceofDiscreteRandomVariables.................257
6.3ContinuousRandomVariables......................268
7SumsofRandomVariables 285
7.1SumsofDiscreteRandomVariables ..................285
7.2SumsofContinuousRandomVariables.................291
8LawofLargeNumbers 305
8.1DiscreteRandomVariables.......................305
8.2ContinuousRandomVariables......................316
v
vi CONTENTS
9CentralLimitTheorem 325
9.1BernoulliTrials..............................325
9.2DiscreteIndependentTrials.......................340
9.3ContinuousIndependentTrials.....................355
10GeneratingFunctions 365
10.1DiscreteDistributions..........................365
10.2BranchingProcesses...........................377
10.3ContinuousDensities...........................394
11MarkovChains 405
11.1Introduction................................405
11.2AbsorbingMarkovChains........................415
11.3ErgodicMarkovChains.........................433
11.4FundamentalLimitTheorem......................447
11.5MeanFirstPassageTime........................452
12RandomWalks 471
12.1RandomWalksinEuclideanSpace...................471
12.2Gambler’sRuin..............................486
12.3ArcSineLaws...............................493
Appendices 499
ANormalDistributionTable........................499
BGalton’sData...............................500
CLifeTable.................................501
Index
503
Preface
ProbabilitytheorybeganinseventeenthcenturyFrancewhenthetwogreatFrench
mathematicians,BlaisePascalandPierredeFermat,correspondedovertwoprob-
lemsfromgamesofchance.ProblemslikethosePascalandFermatsolvedcontinued
toinfluencesuchearlyresearchersasHuygens,Bernoulli,andDeMoivreinestab-
lishingamathematicaltheoryofprobability.Today,probabilitytheoryisawell-
establishedbranchofmathematicsthatfindsapplicationsineveryareaofscholarly
activityfrommusictophysics,andindailyexperiencefromweatherpredictionto
predictingtherisksofnewmedicaltreatments.
Thistextisdesignedforanintroductoryprobabilitycoursetakenbysophomores,
juniors,andseniorsinmathematics,thephysicalandsocialsciences,engineering,
andcomputerscience.Itpresentsathoroughtreatmentofprobabilityideasand
techniquesnecessaryforafirmunderstandingofthesubject.Thetextcanbeused
inavarietyofcourselengths,levels,andareasofemphasis.
Foruseinastandardone-termcourse,inwhichbothdiscreteandcontinuous
probabilityiscovered,studentsshouldhavetakenasaprerequisitetwotermsof
calculus,includinganintroductiontomultipleintegrals.InordertocoverChap-
ter11,whichcontainsmaterialonMarkovchains,someknowledgeofmatrixtheory
isnecessary.
Thetextcanalsobeusedinadiscreteprobabilitycourse.Thematerialhasbeen
organizedinsuchawaythatthediscreteandcontinuousprobabilitydiscussionsare
presentedinaseparate,butparallel,manner.Thisorganizationdispelsanoverly
rigorousorformalviewofprobabilityandoerssomestrongpedagogicalvalue
inthatthediscretediscussionscansometimesservetomotivatethemoreabstract
continuousprobabilitydiscussions.Foruseinadiscreteprobabilitycourse,students
shouldhavetakenonetermofcalculusasaprerequisite.
Verylittlecomputingbackgroundisassumedornecessaryinordertoobtainfull
benefitsfromtheuseofthecomputingmaterialandexamplesinthetext.Allof
theprogramsthatareusedinthetexthavebeenwrittenineachofthelanguages
TrueBASIC,Maple,andMathematica.
ThisbookisontheWebathttp://www.dartmouth.edu/˜chance,andispartof
theChanceproject,whichisdevotedtoprovidingmaterialsforbeginningcoursesin
probabilityandstatistics.Thecomputerprograms,solutionstotheodd-numbered
exercises,andcurrenterrataarealsoavailableatthissite.Instructorsmayobtain
allofthesolutionsbywritingtoeitheroftheauthors,atjlsnell@dartmouth.eduand
cgrinst1@swarthmore.edu.Itisourintentiontoplaceitemsrelatedtothisbookat
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