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1、CHAPTEREIGHTContinuousDistributionsandthePoissonProcessThischapterintroducesthegeneralconceptofcontinuousrandomvariables,focusingontwoexamplesofcontinuousdistributions:theuniformdistributionandtheexponentialdistribution.WethenproceedtostudythePoissonprocess,acontinuoustimecountingprocessthatisrel
2、atedtoboththeuniformandexponentialdistributions.WeconcludethischapterwithbasicapplicationsofthePoissonprocessinqueueingtheory.8.1.ContinuousRandomVariables8.1.1.ProbabilityDistributionsinRThecontinuousroulettewheelinFigure8.IhascircumferenceI.Wespinthewheel,andwhenitstops.theoutcomeistheclockwise
3、distanceX(computedwithinfiniteprecision)fromthe··o··marktothearrow.ThesamplespaceQofthisexperimentconsistsofallrealnumbersintherange[0,I).Assumethatanypointonthecircumferenceofthediskisequallylikelytofacethearrowwhenthediskstops.Whatistheprobabilitypofagivenoutcomex'?Toanswerthisquestion,werecall
4、thatinChapterIwedefinedaprobabilityfunctiontobeanyfunctionthatsatisfiesthefollowingthreerequirements:1.Pr(Q)=I;2.foranyeventE,0.SPr(E).SI;3.forany(finiteorenumerable)collectionBofdisjointevents,Pr(ur;)=LPr(f:).EEBEEBLetS(k)beasetofkdistinctpointsintherange[0,1),andletpbetheprobabilitythatanygiven
5、pointin[0,I)istheoutcomeoftherouletteexperiment.SincetheprobabilityofanyeventisboundedbyI,1888.1CONTINUOUSRANDOMVARIABLESFigure8.1:Acontinuousroulettewheel.Pr(xES(k))=kpS1.\·ecanchooseanynumberkofdistinctpointsintherange[0,I).sowemusthave�/)sIforanyintegerk,whichimpliesthatp=0.Thus,weobservethat
6、inaninfinite'amplespacetheremaybepossibleeventsthathaveprobability0.Takingthecomplementofsuchanevent,weobservethatinaninfinitesamplespacetherecanbeevents\ithprobabilityIthatdonotcorrespondtoallpossibleexperimentaloutcomes.andthustherecanbeeventswithprobabilityIthatare,insomesense.notcertain�lfth
7、eprobabilityofeachpossibleoutcomeofourexperimentis0.howdowedefinetheprobabilityoflargereventswithnonzeroprobability'?Forprobabilitydistributions,,erR,probabilitiesareassignedtointen·aisratherthantoindiidual�tlue:-;.
