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1、ConditionalityandstoppingtimesinprobabilityMarkOsegard,BenSpeidel,MeganSilberhorn,andDickensNyabutiConditionalExpectationConditionalProbabilityDiscrete:ConditionalProbabilityMassFunctionContinuous:ConditionalProbabilityDensityFunctionConditionalExpectationDiscrete:Continuous
2、:Note:ofy.Wewritethisasisafunctioni.e.(ConditionalExpectationFunction)Theorem:Clearly,whenYisdiscrete,WhenYiscontinuous,Proof:ContinuousCaseRecall,ifX,YarejointlycontinuouswithjointpdfDefine:andNote:ContinuousCaseCont.(Fubini’sTheorem)So,Therefore,concludingSummary:WhenYisdi
3、screte,WhenYiscontinuous,ConditionalVarianceDefinitionProofNoteaswell……addinggStoppingtimesStoppingTimesDefinitionApplicationtoProbabilityApplicationsofStoppingTimestootherformulasStoppingTimesBasicDefinition:AStoppingTimeforaprocessdoesexactlythat,ittellstheprocesswhentosto
4、p.Ex)while(x!=4){…}Thestoppingtimeforthiscodefragmentwouldbetheinstancewherexdoesequal4.StoppingtimesinSequencesDefine:SupposewehaveasequenceofRandomVariables(allindependentofeachother)Oursequencethenwouldbe:StoppingTimes:ADiscreteCaseFromourpreviousslidewehavethesequence:Ad
5、iscreteRandomVariableNisastoppingtimeforthissequenceif:{N=n}WherenisindependentofallfollowingitemsinthesequenceIndependenceSummarizingtheideaofstoppingtimeswithRandomVariablesweseethatthedecisionmadetostopthesequenceatRandomVariableNdependssolelyonthevaluesofthesequenceBecau
6、seofthis,wethencanseethatNisindependentofallremainingvaluesApplicationsofStoppingTimesDoesStoppingTimesaffectexpectation?No!Considerthisstatement:Thisformula,theformulausedforConditionalExpectationdoesremainunchangedApplyingStoppingTimesForanexampleofhowtousestoppingtimestos
7、olveaproblem,wewillnowintroducetoyouWald’sEquation…Wald’sEquationPropositionIf{X1,X2,X3,…}areindependentidenticallydistributed(iid)randomvariableshavingafiniteexpectationE[X],andNisastoppingtimeforthesequencehavingfiniteexpectationE[N],then:Wald’sProofLetN1=Nrepresentthestop
8、pingtimeforthesequence{X1,X2,…,XN1}LetN2=thestoppingtimeforthesequence{X(N1
