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首页 随机过程, ross版,经典教材

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随机过程, ross版,经典教材

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IntroductiontoProbabilityModelsNinthEditionThispageintentionallyleftblankIntroductiontoProbabilityModelsNinthEditionSheldonMRossUniversityofCaliforniaBerkeley,CaliforniaAMSTERDAM•BOSTON•HEIDELBERG•LONDONNEWYORK•OXFORD•PARIS•SANDIEGOSANFRANCISCO•SINGAPORE•SYDNEY•TOKYOAcademicPressisanimprintofElsevierAcquisitionsEditorTomSingerProjectManagerSarahMHajdukMarketingManagersLindaBeattie,LeahAckersonCoverDesignEricDeCiccoCompositionVTEXCoverPrinterPhoenixColorInteriorPrinterTheMapleVailBookManufacturingGroupAcademicPressisanimprintofElsevierCorporateDrive,Suite,Burlington,MA,USABStreet,Suite,SanDiego,California,USATheobald’sRoad,LondonWCXRR,UKThisbookisprintedonacidfreepaper©∞Copyright©,ElsevierIncAllrightsreservedNopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,includingphotocopy,recording,oranyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisherPermissionsmaybesoughtdirectlyfromElsevier’sScienceTechnologyRightsDepartmentinOxford,UK:phone:(),fax:(),Email:permissionselseviercomYoumayalsocompleteyourrequestonlineviatheElsevierhomepage(http:elseviercom),byselecting“SupportContact”then“CopyrightandPermission”andthen“ObtainingPermissions”LibraryofCongressCataloginginPublicationDataApplicationSubmittedBritishLibraryCataloguinginPublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibraryISBN:ISBN:ForinformationonallAcademicPresspublicationsvisitourWebsiteatwwwbookselseviercomPrintedintheUnitedStatesofAmericaContentsPrefacexiiiIntroductiontoProbabilityTheoryIntroductionSampleSpaceandEventsProbabilitiesDefinedonEventsConditionalProbabilitiesIndependentEventsBayes’FormulaExercisesReferencesRandomVariablesRandomVariablesDiscreteRandomVariablesTheBernoulliRandomVariableTheBinomialRandomVariableTheGeometricRandomVariableThePoissonRandomVariableContinuousRandomVariablesTheUniformRandomVariableExponentialRandomVariablesGammaRandomVariablesNormalRandomVariablesvviContentsExpectationofaRandomVariableTheDiscreteCaseTheContinuousCaseExpectationofaFunctionofaRandomVariableJointlyDistributedRandomVariablesJointDistributionFunctionsIndependentRandomVariablesCovarianceandVarianceofSumsofRandomVariablesJointProbabilityDistributionofFunctionsofRandomVariablesMomentGeneratingFunctionsTheJointDistributionoftheSampleMeanandSampleVariancefromaNormalPopulationLimitTheoremsStochasticProcessesExercisesReferencesConditionalProbabilityandConditionalExpectationIntroductionTheDiscreteCaseTheContinuousCaseComputingExpectationsbyConditioningComputingVariancesbyConditioningComputingProbabilitiesbyConditioningSomeApplicationsAListModelARandomGraphUniformPriors,Polya’sUrnModel,andBose–EinsteinStatisticsMeanTimeforPatternsThekRecordValuesofDiscreteRandomVariablesAnIdentityforCompoundRandomVariablesPoissonCompoundingDistributionBinomialCompoundingDistributionACompoundingDistributionRelatedtotheNegativeBinomialExercisesContentsviiMarkovChainsIntroductionChapman–KolmogorovEquationsClassificationofStatesLimitingProbabilitiesSomeApplicationsTheGambler’sRuinProblemAModelforAlgorithmicEfficiencyUsingaRandomWalktoAnalyzeaProbabilisticAlgorithmfortheSatisfiabilityProblemMeanTimeSpentinTransientStatesBranchingProcessesTimeReversibleMarkovChainsMarkovChainMonteCarloMethodsMarkovDecisionProcessesHiddenMarkovChainsPredictingtheStatesExercisesReferencesTheExponentialDistributionandthePoissonProcessIntroductionTheExponentialDistributionDefinitionPropertiesoftheExponentialDistributionFurtherPropertiesoftheExponentialDistributionConvolutionsofExponentialRandomVariablesThePoissonProcessCountingProcessesDefinitionofthePoissonProcessInterarrivalandWaitingTimeDistributionsFurtherPropertiesofPoissonProcessesConditionalDistributionoftheArrivalTimesEstimatingSoftwareReliabilityGeneralizationsofthePoissonProcessNonhomogeneousPoissonProcessCompoundPoissonProcessConditionalorMixedPoissonProcessesviiiContentsExercisesReferencesContinuousTimeMarkovChainsIntroductionContinuousTimeMarkovChainsBirthandDeathProcessesTheTransitionProbabilityFunctionPij(t)LimitingProbabilitiesTimeReversibilityUniformizationComputingtheTransitionProbabilitiesExercisesReferencesRenewalTheoryandItsApplicationsIntroductionDistributionofN(t)LimitTheoremsandTheirApplicationsRenewalRewardProcessesRegenerativeProcessesAlternatingRenewalProcessesSemiMarkovProcessesTheInspectionParadoxComputingtheRenewalFunctionApplicationstoPatternsPatternsofDiscreteRandomVariablesTheExpectedTimetoaMaximalRunofDistinctValuesIncreasingRunsofContinuousRandomVariablesTheInsuranceRuinProblemExercisesReferencesQueueingTheoryIntroductionPreliminariesCostEquationsSteadyStateProbabilitiesContentsixExponentialModelsASingleServerExponentialQueueingSystemASingleServerExponentialQueueingSystemHavingFiniteCapacityAShoeshineShopAQueueingSystemwithBulkServiceNetworkofQueuesOpenSystemsClosedSystemsTheSystemMGPreliminaries:WorkandAnotherCostIdentityApplicationofWorktoMGBusyPeriodsVariationsontheMGTheMGwithRandomSizedBatchArrivalsPriorityQueuesAnMGOptimizationExampleTheMGQueuewithServerBreakdownTheModelGMTheGMBusyandIdlePeriodsAFiniteSourceModelMultiserverQueuesErlang’sLossSystemTheMMkQueueTheGMkQueueTheMGkQueueExercisesReferencesReliabilityTheoryIntroductionStructureFunctionsMinimalPathandMinimalCutSetsReliabilityofSystemsofIndependentComponentsBoundsontheReliabilityFunctionMethodofInclusionandExclusionSecondMethodforObtainingBoundsonr(p)SystemLifeasaFunctionofComponentLivesExpectedSystemLifetimeAnUpperBoundontheExpectedLifeofaParallelSystemxContentsSystemswithRepairASeriesModelwithSuspendedAnimationExercisesReferencesBrownianMotionandStationaryProcessesBrownianMotionHittingTimes,MaximumVariable,andtheGambler’sRuinProblemVariationsonBrownianMotionBrownianMotionwithDriftGeometricBrownianMotionPricingStockOptionsAnExampleinOptionsPricingTheArbitrageTheoremTheBlackScholesOptionPricingFormulaWhiteNoiseGaussianProcessesStationaryandWeaklyStationaryProcessesHarmonicAnalysisofWeaklyStationaryProcessesExercisesReferencesSimulationIntroductionGeneralTechniquesforSimulatingContinuousRandomVariablesTheInverseTransformationMethodTheRejectionMethodTheHazardRateMethodSpecialTechniquesforSimulatingContinuousRandomVariablesTheNormalDistributionTheGammaDistributionTheChiSquaredDistributionTheBeta(n,m)DistributionTheExponentialDistributionTheVonNeumannAlgorithmSimulatingfromDiscreteDistributionsTheAliasMethodContentsxiStochasticProcessesSimulatingaNonhomogeneousPoissonProcessSimulatingaTwoDimensionalPoissonProcessVarianceReductionTechniquesUseofAntitheticVariablesVarianceReductionbyConditioningControlVariatesImportanceSamplingDeterminingtheNumberofRunsCouplingfromthePastExercisesReferencesAppendix:SolutionstoStarredExercisesIndexThispageintentionallyleftblankPrefaceThistextisintendedasanintroductiontoelementaryprobabilitytheoryandstochasticprocessesItisparticularlywellsuitedforthosewantingtoseehowprobabilitytheorycanbeappliedtothestudyofphenomenainfieldssuchasengineering,computerscience,managementscience,thephysicalandsocialsciences,andoperationsresearchItisgenerallyfeltthattherearetwoapproachestothestudyofprobabilitytheoryOneapproachisheuristicandnonrigorousandattemptstodevelopinthestudentanintuitivefeelforthesubjectwhichenableshimorherto“thinkprobabilistically”TheotherapproachattemptsarigorousdevelopmentofprobabilitybyusingthetoolsofmeasuretheoryItisthefirstapproachthatisemployedinthistextHowever,becauseitisextremelyimportantinbothunderstandingandapplyingprobabilitytheorytobeableto“thinkprobabilistically,”thistextshouldalsobeusefultostudentsinterestedprimarilyinthesecondapproachNewtoThisEditionThenintheditioncontainsthefollowingnewsections•SectionisconcernedwithcompoundrandomvariablesoftheformSN=∑Ni=Xi,whereNisindependentofthesequenceofindependentandidenticallydistributedrandomvariablesXi,i�Itstartsbyderivingageneralidentityconcerningcompoundrandomvariables,aswellasacorollaryofthatidentityinthecasewheretheXiarepositiveandintegervaluedThecorollaryisthenusedinsubsequentsubsectionstoobtainrecursiveformulasfortheprobabilitymassfunctionofSN,whenNisaPoissondistribution(Subsection),abinomialdistribution(Subsection),oranegativebinomialdistribution(Subsection)xiiixivPreface•SectiondealswithhiddenMarkovchainsThesemodelssupposethatarandomsignalisemittedeachtimeaMarkovchainentersastate,withthedistributionofthesignaldependingonthestateenteredTheMarkovchainishiddeninthesensethatitissupposedthatonlythesignalsandnottheunderlyingstatesofthechainareobservableAspartofouranalysisofthesemodelswepresent,inSubsection,theViterbialgorithmfordeterminingthemostprobablesequenceoffirstnstates,giventhefirstnsignals•SectionanalyzesthePoissonarrivalsingleserverqueueundertheassumptionthattheworkingserverwillrandomlybreakdownandneedrepairThereisalsonewmaterialinalmostallchaptersSomeofthemoresignificantadditionsbeingthefollowing•Example,whichisconcernedwiththeexpectednumberofnormalcellsthatsurviveuntilallcancercellshavebeenkilledTheexamplesupposesthateachcellhasaweight,andtheprobabilitythatagivensurvivingcellisthenextcellkilledisproportionaltoitsweight•AnewapproachbasedontimesamplingofaPoissonprocessispresentedinSubsectionforderivingtheprobabilitymassfunctionofthenumberofeventsofanonhomogeneousPoissonprocessthatoccurinanyspecifiedtimeinterval•ThereisadditionalmaterialinSectionconcerningtheMMqueueAmongotherthings,wederivetheconditionaldistributionofthenumberofcustomersoriginallyfoundinthesystembyacustomerwhospendsatimetinthesystembeforedeparting(TheconditionaldistributionisPoisson)InExample,weillustratetheinspectionparadox,byobtainingtheprobabilitydistributionofthenumberinthesystemasseenbythefirstarrivalaftersomespecifiedtimeCourseIdeally,thistextwouldbeusedinaoneyearcourseinprobabilitymodelsOtherpossiblecourseswouldbeaonesemestercourseinintroductoryprobabilitytheory(involvingChapters–andpartsofothers)oracourseinelementarystochasticprocessesThetextbookisdesignedtobeflexibleenoughtobeusedinavarietyofpossiblecoursesForexample,IhaveusedChaptersand,withsmatteringsfromChaptersand,asthebasisofanintroductorycourseinqueueingtheoryPrefacexvExamplesandExercisesManyexamplesareworkedoutthroughoutthetext,andtherearealsoalargenumberofexercisestobesolvedbystudentsMorethanoftheseexerciseshavebeenstarredandtheirsolutionsprovidedattheendofthetextThesestarredproblemscanbeusedforindependentstudyandtestpreparationAnInstructor’sManual,containingsolutionstoallexercises,isavailablefreetoinstructorswhoadoptthebookforclassOrganizationChaptersanddealwithbasicideasofprobabilitytheoryInChapteranaxiomaticframeworkispresented,whileinChaptertheimportantconceptofarandomvariableisintroducedSubsectiongivesasimplederivationofthejointdistributionofthesamplemeanandsamplevarianceofanormaldatasampleChapterisconcernedwiththesubjectmatterofconditionalprobabilityandconditionalexpectation“Conditioning”isoneofthekeytoolsofprobabilitytheory,anditisstressedthroughoutthebookWhenproperlyused,conditioningoftenenablesustoeasilysolveproblemsthatatfirstglanceseemquitedifficultThefinalsectionofthischapterpresentsapplicationsto()acomputerlistproblem,()arandomgraph,and()thePolyaurnmodelanditsrelationtotheBoseEinsteindistributionSubsectionpresentskrecordvaluesandthesurprisingIgnatov’stheoremInChapterwecomeintocontactwithourfirstrandom,orstochastic,process,knownasaMarkovchain,whichiswidelyapplicabletothestudyofmanyrealworldphenomenaApplicationstogeneticsandproductionprocessesarepresentedTheconceptoftimereversibilityisintroducedanditsusefulnessillustratedSubsectionpresentsananalysis,basedonrandomwalktheory,ofaprobabilisticalgorithmforthesatisfiabilityproblemSectiondealswiththemeantimesspentintransientstatesbyaMarkovchainSectionintroducesMarkovchainMonteCarlomethodsInthefinalsectionweconsideramodelforoptimallymakingdecisionsknownasaMarkoviandecisionprocessInChapterweareconcernedwithatypeofstochasticprocessknownasacountingprocessInparticular,westudyakindofcountingprocessknownasaPoissonprocessTheintimaterelationshipbetweenthisprocessandtheexponentialdistributionisdiscussedNewderivationsforthePoissonandnonhomogeneousPoissonprocessesarediscussedExamplesrelatingtoanalyzinggreedyalgorithms,minimizinghighwayencounters,collectingcoupons,andtrackingtheAIDSvirus,aswellasmaterialoncompoundPoissonprocesses,areincludedxviPrefaceinthischapterSubsectiongivesasimplederivationoftheconvolutionofexponentialrandomvariablesChapterconsidersMarkovchainsincontinuoustimewithanemphasisonbirthanddeathmodelsTimereversibilityisshowntobeausefulconcept,asitisinthestudyofdiscretetimeMarkovchainsSectionpresentsthecomputationallyimportanttechniqueofuniformizationChapter,therenewaltheorychapter,isconcernedwithatypeofcountingprocessmoregeneralthanthePoissonBymakinguseofrenewalrewardprocesses,limitingresultsareobtainedandappliedtovariousfieldsSectionpresentsnewresultsconcerningthedistributionoftimeuntilacertainpatternoccurswhenasequenceofindependentandidenticallydistributedrandomvariablesisobservedInSubsection,weshowhowrenewaltheorycanbeusedtoderiveboththemeanandthevarianceofthelengthoftimeuntilaspecifiedpatternappears,aswellasthemeantimeuntiloneofafinitenumberofspecifiedpatternsappearsInSubsection,wesupposethattherandomvariablesareequallylikelytotakeonanyofmpossiblevalues,andcomputeanexpressionforthemeantimeuntilarunofmdistinctvaluesoccursInSubsection,wesupposetherandomvariablesarecontinuousandderiveanexpressionforthemeantimeuntilarunofmconsecutiveincreasingvaluesoccursChapterdealswithqueueing,orwaitingline,theoryAftersomepreliminariesdealingwithbasiccostidentitiesandtypesoflimitingprobabilities,weconsiderexponentialqueueingmodelsandshowhowsuchmodelscanbeanalyzedIncludedinthemodelswestudyistheimportantclassknownasanetworkofqueuesWethenstudymodelsinwhichsomeofthedistributionsareallowedtobearbitraryIncludedareSubsectiondealingwithanoptimizationproblemconcerningasingleserver,generalservicetimequeue,andSection,concernedwithasingleserver,generalservicetimequeueinwhichthearrivalsourceisafinitenumberofpotentialusersChapterisconcernedwithreliabilitytheoryThischapterwillprobablybeofgreatestinteresttotheengineerandoperationsresearcherSubsectionillustratesamethodfordetermininganupperboundfortheexpectedlifeofaparallelsystemofnotnecessarilyindependentcomponentsand()analyzingaseriesstructurereliabilitymodelinwhichcomponentsenterastateofsuspendedanimationwhenoneoftheircohortsfailsChapterisconcernedwithBrownianmotionanditsapplicationsThetheoryofoptionspricingisdiscussedAlso,thearbitragetheoremispresentedanditsrelationshiptothedualitytheoremoflinearprogramisindicatedWeshowhowthearbitragetheoremleadstotheBlack–ScholesoptionpricingformulaChapterdealswithsimulation,apowerfultoolforanalyzingstochasticmodelsthatareanalyticallyintractableMethodsforgeneratingthevaluesofarbitrarilydistributedrandomvariablesarediscussed,asarevariancereductionmethodsforincreasingtheefficiencyofthesimulationSubsectionintroducesthePrefacexviiimp

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