概率论中条件期望与停时

ConditionalityandstoppingtimesinprobabilityMarkOsegard,BenSpeidel,MeganSilberhorn,andDickensNyabutiConditionalExpectationConditionalProbabilityDiscrete:ConditionalProbabilityMassFunctionyYPyYxXP,yYxXP|Continuous:ConditionalProbabilityDensityFunction)(),(:)|(|yfyxfyxfXYXConditionalExpectationDiscrete:dxyxxfyYXEYX),(||Continuous:xyYxXxPyYXE||Note:yYXEy|ofy.WewritethisasYXE|isafunctioni.e.yYXEyYXE||(ConditionalExpectationFunction)Theorem:YXEEXE|Clearly,whenYisdiscrete,yyYPyYXE|WhenYiscontinuous,dyyfyYXEY|Proof:ContinuousCaseRecall,ifX,Yarejointlycontinuouswithjointpdfyxf,Define:yfyxfYXfYYX,||dxYXxfyYXEYX|||andNote:dyyfdxyxxfdyyfyYXEYYXY|||dxdyyfyxxfYYX||dxdyyfyfyxfxYY,ContinuousCaseCont.dxdyyxxf,dydxyxxf,(Fubini’sTheorem)dydxyxfx,dyyxf,So,XEdxxxfXTherefore,concludingYXEEXE|Summary:WhenYisdiscrete,yyYPyYXE|WhenYiscontinuous,dyyfyYXEY|ConditionalVarianceDefinitionYYXEXEYXVar||)|(222||YXEYXEYXEVarYXVarEXVar||ProofYXEEXE|using22222222|||||||||YXEEXEYXEEYXEEYXEYXEEYXVarEYXEYXEYXVarsidesbothofnsexpectatiotakingsinceNoteaswell…22|||YXEEYXEEYXEVar22|XEYXEE…addingYXEVarYXVarEXVarXVarXEXEYXEVarYXVarE||||22thatshownvewe'ThusgStoppingtimesStoppingTimesDefinitionApplicationtoProbabilityApplicationsofStoppingTimestootherformulasStoppingTimesBasicDefinition:AStoppingTimeforaprocessdoesexactlythat,ittellstheprocesswhentostop.Ex)while(x!=4){…}Thestoppingtimeforthiscodefragmentwouldbetheinstancewherexdoesequal4.StoppingtimesinSequencesDefine:SupposewehaveasequenceofRandomVariables(allindependentofeachother)Oursequencethenwouldbe:,...,,321XXXStoppingTimes:ADiscreteCaseFromourpreviousslidewehavethesequence:AdiscreteRandomVariableNisastoppingtimeforthissequenceif:{N=n}Wherenisindependentofallfollowingitemsinthesequence,...,,321XXX,...,21nnXXIndependenceSummarizingtheideaofstoppingtimeswithRandomVariablesweseethatthedecisionmadetostopthesequenceatRandomVariableNdependssolelyonthevaluesofthesequenceBecauseofthis,wethencanseethatNisindependentofallremainingvaluesnXXX,...,,21nmXm,ApplicationsofStoppingTimesDoesStoppingTimesaffectexpectation?No!Considerthisstatement:Thisformula,theformulausedforConditionalExpectationdoesremainunchangedNiiXS1][][][XENESEApplyingStoppingTimesForanexampleofhowtousestoppingtimestosolveaproblem,wewillnowintroducetoyouWald’sEquation…][][][1XENEXENiiWald’sEquationPropositionIf{X1,X2,X3,…}areindependentidenticallydistributed(iid)randomvariableshavingafiniteexpectationE[X],andNisastoppingtimeforthesequencehavingfiniteexpectationE[N],then:][][1XENEXENiiWald’sProofLetN1=Nrepresentthestoppingtimeforthesequence{X1,X2,…,XN1}LetN2=thestoppingtimeforthesequence{X(N1+1),X(N1+2),…,X(N1+N2)}LetN3=thestoppingtimeforthesequence{X(N1+N2+1),X(N1+N2+2),…,X(N1+N2+N3)}Wald’sProof...Wecannowdefinethesequenceofstoppingtimesaswhere{Ni}clearlyrepresents,andseethesequenceisiid1i}{iN...,N,N,321NWald’sProof…miSSSS.....21Ifwedefineasequence{Si}as,where,12112112111......m21S,...,,}{NiNNNiNNNNNNiiiiimmXXSXSSNote:{Si}areiidWald’sProof…withcommonmeanE[Xi]=E[X]{N1+N2+...+Nm}whichareiidbecausetheXi’sare.ofconsistwillmiSSSS.....21Wald’sProof…BytheStrongLawofLargeNumbers,{N1+N2+...+Nm}{S1+S2+...+Sm}=E[X]mlimWald’sProof…mNNNmSSSmm..........2121AlsomletSENEXEConcludingmSoasweletWhichcanbemanipulatedintoourpreposition:][][][1XENEXESENiiNESEXEMinersProblemSampleConditionalandStoppingtimesinprobabilityproblemTheproblemAmineristrappedinaroomcontainingthreedoors.Dooroneleadstoatunnelthatreturnstothesameroomafter4days;doortwoleadstoatunnelthatreturnstothesameroomafter7days;doorthreeleadstofreedomaftera3dayjourney.Iftheminerisatalltimesequallylikelytochooseanyofthedoors,findtheexpectedvalueandthevarianceofthetimeittakestheminertobecomefreeExpectedValueUsingWald’sEquation:3|min3,7,4iiXiNNXtimestoppingtheContinue….time.stoppingthedenotetoNtosetbewillNumbertrialitsoccurs,eventthisoncesuccessdenotes3doorselecting31yProbabilitawithondistributigeometricaisN,Continue….3311NEp1isondistributiGeometricaofvalueexpectedThe:RecallExpectedvalueConclusiondays14infreedomattaintoexpectedisminertheaverage,onThusgetweEquation,sWald'usingngSubstitutidaysXEXENEXEi143143VarianceniiniinNXEnNXEnNXENXEVarNXVarEXVar11|||||)(:SolutionequationtheusingContinue…..niniiiiiiinNXEnNXPnNXPnNXPnNxXxPnNXE2113||33|77|44||Continue…..25211|2521131211|||111NNXEnnnNXEnNXEnNXEiniiniiContinue…..6313111412125211|)(222PPNVarNVarNVarNXEVarxVarabaXVarthenusingThus