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1、9/28/12Probability•Probabilityallowsustoquan5fystatementsaboutthechanceofaneventtakingplace.SimulationForexample‐Flipafaircoin1.What’sthechanceitlandsheads?2.Flipit45mes,whatpropor5onofheadsdoyouexpect?3.Willyougetexactlythatpropor5on?4.Whathappenswhenyouflipthecoin10005mes?4Flips4Flips•In4flips,weca
2、nget0,1,2,3,or4Headsandsothe•Wecanthinkofthepropor5onofHeadsin4flipsaspropor5onofHeadscanbe:0,0.25,0.5,0.75,or1asta5s5cbecauseitsummarizesdata•Weexpectthepropor5ontobe0.5•But,apropor5onof0.25isquitelikely:•No5cethatitisarandomquan5ty–ittakeson5Thereare16possiblewaysfor4tossestoland,e.g.possiblevalue
3、s,eachwithsomeprobabilityHHHH,HHHT,HHTH,…value0.000.250.500.751.00Eachisequallylikely,sothechanceofanypar5cularchance1/164/166/164/161/16sequenceofHsandTsis1/16Sochanceof0.25propor5onis4/16HTTT,THTT,TTHT,TTTH19/28/121,000Flips1,000Flips•Whenweflipthecoin1,0005mes,wecangeta•Withsomeadvancedmathtools,
4、wecanfigurethismanydifferentpossiblepropor5onsofHeads,i.e.0,out.0.001,0.002,0.003,…,0.998,0.999,1.000•Butwecanalsogetagoodideausingasimula5on.•Inoursimula5onwewillassumethatthechanceof•It’shighlyunlikelythatwewouldget0fortheHeadsis0.5andfindoutwhatthepossiblevaluesforpropor5on–howunlikely?thepropor5on
5、ofheadsin1,000flipslookslike•Ifweweretocarryoutanexperimentwithacoin•Whatdoesthedistribu5onofthepropor5onofandgetapar5cularpropor5on,say0.37,thenweheadsin1000flipslooklike?couldusethissimula5onstudytohelpusunderstandtheresultsofourexperiment.Let’sTryItLet’sGeneralize29/28/12Beforeweconsidertheroletha
6、tsimulationcanplayinhelpingAstatisticisoftenjustafunctionofarandomsample,forusunderstandstatistics,let’stakeastepbackandthinkexamplethesamplemean,the95thquantile,orthesampleaboutthebigpicture.proportion.WecanthinkofprobabilitytheoryascomplimentarytoStatisticsareoftenusedasestimatorsofquantitiesofin
7、tereststatisticalinference.aboutthedistribution,calledparameters.Statisticsarerandomvariables(sincetheydependonthesample);Probabilityparametersarenot.Insimplecases,wecanstudythesamplingdistributionoftheDist
