Grinstead and Snell’s Introduction to Probability

1 GrinsteadandSnell'sIntroductionto ProbabilityTheCHANCEP roject1 Versiondated4 July20061 Copyright (C) a versionof GrinsteadandSnell's`Introductionto Probability , 2ndedition',publishedby theAmericanMathematicalSo-ciety, Copyright (C) freelyredistributableunderthetermsof ourwivesandin DiscreteProbability DiscreteProbabilities.. Distributions..182 ContinuousProbability ContinuousProbabilities.. Functions..553 .. ing.. 1204 .. 1755 Distributions.. Densities.. 2056 .. DiscreteRandomVariables.. 2687 Sumsof DiscreteRandomVariables.

2 ContinuousRandomVariables.. 2918 Law of .. 316vviCONTENTS9 .. Trials.. Trials.. 35610 .. 39311 Markov .. Chains.. Chains.. 45212 EuclideanSpace.. 'sRuin.. 493 Appendices499 Index503 PrefaceProbability theorybeganin seventeenth centuryFrancewhenthetwo greatFrenchmathematicians,BlaisePascalan dPierredeFermat,correspondedover two prob-lemsfromgamesof thosePascalandFermatsolved continuedto in uencesuch earlyresearchersas Huygens,Bernoulli,andDeMoivrein estab-lishinga mathematicaltheoryof Probability .

3 Today, Probability theoryis a well-establishedbranch of mathematicsthat ndsapplicationsin everyareaof scholarlyactivity frommusicto physics,andin dailyexperiencefromweatherpredictiontopr edictingtherisksof designedforan introductoryprobability coursetaken by sophomores,juniors,andseniorsin mathematics,thephysicalandsocialsciences ,engineering, presents a thoroughtreatment of Probability ideasandtechniquesnecessaryfora rmunderstandingof usedin a variety of courselengths,levels,andareasof usein a standardone-termcourse,in which bothdiscreteandcontinuousprobability is covered,students shouldhave takenas a prerequisitetwo termsofcalculus,includinganintroductiont o orderto cover Chap-ter11,which containsmaterialonMarkov chains,someknowledgeof matrixtheoryis usedin a discreteprobability such a way thatthediscreteandcontinuousprobability discussionsarepresentedin a separate,butparallel.

4 Probability ando erssomestrongpedagogicalvaluein thatthediscretediscussionscansometimesse rve to motivatethemoreabstractcontinuousprobabi lity usein a discreteprobability course,studentsshouldhave taken onetermof calculusas a assumedor necessaryin orderto obtainfullbene tsfromtheuseof thecomputingmaterialandexamplesin thetexthave beenwrittenin each of thelanguagesTrueBASIC,Maple, distributedontheWeb as partof theChanceProject,which is de-votedto providingmaterialsforbeginningcoursesin Probability ,solutionsto theodd-numberedexercises,andcurrent errataarealsoavailableat obtainallof thesolutionsby writingtoeitherof theauthors,at rigorandemphasis: Probability is a wonderfullyintuitive andapplicable eldof have triednotto spoilitsbeauty by presentingtoo ,we have triedto developthekey ideasin a somewhatleisurelystyle,to providea variety of interestingapplicationsto Probability , andtoshow someof thenonintuitive examplesthatmake Probability such a.

5 Thereareover 600exercisesin thetextprovidingplenty of oppor-tunity forpracticingskillsanddevelopinga soundunderstandingof be donewithandwithouttheuseof acomputerandmoretheoreticalexercisesto improve theunderstandingof cultexercisesareindicatedby solutionmanualforallof theexercisesis availableto remarks:Introductoryprobability is a subjectin which thefunda-mentalideasarestillcloselytiedt o thoseof thefoundersof thisreason,therearenumeroushistoricalcom ments in thetext,especiallyas theydealwiththedevelopment of useof computerprograms.

6 Probability theorymakes predictionsaboutexperiments , it lendsitselfbeautifullyto theuseof computersas a mathematicaltool to thetextthecomputeris utilizedin ,it providesa labora-torywherechanceexperiments canbe simulatedandthestudents cangeta feelingforthevariety of such thecomputerin Probability hasbeenalreadybeautifullyillustratedby WilliamFellerin thesecondeditionof hisfamoustextAn Introductionto ProbabilityTheoryandItsApplications(NewY ork:Wiley, 1950).In thepreface,Fellerwroteabouthistreatment of uctuationin cointossing:\Theresultsaresoamazingandso atvariancewithcommonintuitionthateven sophisticatedcolleaguesdoubtedthatcoinsa ctuallymisbehave as a simulatedexperiment is thereforeincluded.

7 "In additionto providinga laboratoryforthestudent, thecomputeris a powerfulaidin understandingbasicresultsof Probability theory. For example,thegraphicalillustrationof theapproximationof thestandardizedbinomialdistributionsto thenormalcurve is a moreconvincingdemonstrationof theCentralLimitTheoremthanmany of theformalproofsof , thecomputerallowsthestudent tosolve problemsthatdonotlendthemselves to closed-formformulassuch as waitingtimesin ,theintroductionof thecomputerchangestheway in which we lookat many problemsin Probability . For example,beingableto calculateexactbinomialprobabilitiesforex periments upto 1000trialschangestheway we Probability texttoday owes a greatdebtto WilliamFeller,whotaught us allhow to make Probability comealive as a youPREFACEix ndanexample,anapplication,or anexercisethatyoureallylike, it probablyhaditsoriginin Feller'sclassictext.

8 An Introductionto Probability areindebtedto many peoplefortheirhelpin Markov Chainspresentedin thebookwas developedby JohnKemeny a silent co-authorforthematerialoncontinuousproba bility in anearlierversionof pagesof comments of thesecomments wereverythought-provoking;in addition,theyprovideda student'sperspective thisversionof thebookhave theirgenesisin providedextensive helpwiththetypesettingandthe thecomputerprogramsin , we thanktheAmericanMathematicalSociety, andin particularSergeiGelfandandJohnEwing,fort heirinterestin thisbook;theirhelpin its production;andtheirwillingnessto make DiscreteProbabilitiesProbabilityIn thischapter,we shall rstconsiderchanceexperiments witha nitenumber ofpossibleoutcomes!

9 1,!2, .. ,!n. Forexample,we rolla dieandthepossibleoutcomesare1, 2, 3, 4, 5, 6 correspondingto tossa coinwithpossibleoutcomesH (heads)andT (tails).It is frequentlyusefulto be ableto referto anoutcomeof anexperiment. Forexample,we might want to writethemathematicalexpressionwhich gives thesumof fourrollsof a dothis,we couldletXi,i= 1;2;3;4;represent thevaluesof theoutcomesof thefourrolls,andthenwe couldwritetheexpressionX1+X2+X3+X4forthe sumof 's arecalledrandomvariables. A randomvari-ableis simplyanexpressionwhosevalueis theoutcomeof a in thecaseof othertypes of variablesin mathematics,randomvariablescantake ondi erent therandomvariablewhich represents therollof shallassignprobabilitiesto thepossibleoutcomesof dothisbyassigningto each outcome!

10 Ja nonnegative numberm(!j) in such a way thatm(!1) +m(!2) + +m(!6) = 1:Thefunctionm(!j) is calledthedistributionfunctionof thecaseof therollof thediewe wouldassignequalprobabilitiesor probabilities1/6to each of of probabilities,onecouldwriteP(X 4) =2312 CHAPTER1. DISCRETEPROBABILITYDISTRIBUTIONSto meanthattheprobability is 2=3 thata rollof a diewillhave a valuewhich therandomvariablewhich represents thetossof a thiscase,therearetwo possibleoutcomes,which we canlabel as H havereasonto suspectthatthecoincomesuponeway moreoftenthantheotherway,it is naturalto assigntheprobability of 1/2to each of thetwo bothof theabove experiments,each outcomeis assignedan thecasein example,if a drugis foundtobe e ective 30percent of thetimeit is used,we might assigna Probability .