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1、ConditionalityandstoppingtimesinprobabilityMarkOsegard,BenSpeidel,MeganSilberhorn,andDickensNyabutiConditionalExpectationConditionalProbabilityDiscrete:ConditionalProbabilityMassFunctionContinuous:ConditionalProbabilityDensityFunctionConditionalExpectationDiscrete
2、:Continuous:Note:ofy.Wewritethisasisafunctioni.e.(ConditionalExpectationFunction)Theorem:Clearly,whenYisdiscrete,WhenYiscontinuous,Proof:ContinuousCaseRecall,ifX,YarejointlycontinuouswithjointpdfDefine:andNote:ContinuousCaseCont.(Fubini’sTheorem)So,Therefore,concl
3、udingSummary:WhenYisdiscrete,WhenYiscontinuous,ConditionalVarianceDefinitionProofNoteaswell……addinggStoppingtimesStoppingTimesDefinitionApplicationtoProbabilityApplicationsofStoppingTimestootherformulasStoppingTimesBasicDefinition:AStoppingTimeforaprocessdoesexact
4、lythat,ittellstheprocesswhentostop.Ex)while(x!=4){…}Thestoppingtimeforthiscodefragmentwouldbetheinstancewherexdoesequal4.StoppingtimesinSequencesDefine:SupposewehaveasequenceofRandomVariables(allindependentofeachother)Oursequencethenwouldbe:StoppingTimes:ADiscrete
5、CaseFromourpreviousslidewehavethesequence:AdiscreteRandomVariableNisastoppingtimeforthissequenceif:{N=n}WherenisindependentofallfollowingitemsinthesequenceIndependenceSummarizingtheideaofstoppingtimeswithRandomVariablesweseethatthedecisionmadetostopthesequenceatRa
6、ndomVariableNdependssolelyonthevaluesofthesequenceBecauseofthis,wethencanseethatNisindependentofallremainingvaluesApplicationsofStoppingTimesDoesStoppingTimesaffectexpectation?No!Considerthisstatement:Thisformula,theformulausedforConditionalExpectationdoesremainun
7、changedApplyingStoppingTimesForanexampleofhowtousestoppingtimestosolveaproblem,wewillnowintroducetoyouWald’sEquation…Wald’sEquationPropositionIf{X1,X2,X3,…}areindependentidenticallydistributed(iid)randomvariableshavingafiniteexpectationE[X],andNisastoppingtimefort
8、hesequencehavingfiniteexpectationE[N],then:Wald’sProofLetN1=Nrepresentthestoppingtimeforthesequence{X1,X2,…,XN1}LetN2=thestoppingtimeforthesequence{X(N1
