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Chapter 4 Chance experiments（第2页）

Thislumpsistsofagigaoms，alladepely.Fo:atsomeradecaysbyemittingaparticle.Wedonotkhiswillbe，butsiheatomsinthelumpde22years，thecethat　thispartidecayseriodis50%.Supposeithaserfiveyears:atthattime，itisjustoomintheresiduallumpof　210Pb，sotheceitdecayswithinafurther22yearsisagain50%.Andifithashehesameapplies，andsoon.

Itturnsoutthattheonlywaythisiswheimeuntilagivenatomdecayshaswhatisknooialdistributioyhasthegeneralshapeshowniheheightofthecurvefallingatatrate.Asimilarbadappliestoroadats:ifnonehasothepastweek，thatseemsuoaffecttheathefuture，soweexpectthetimetowaitforaroadatalsotofollooialdistribution.

ThisdistributionisiiedupwiththePoissondistributiohingsareessentiallyatrandom–fiashesoflightninginastorm，spoioion，thearrivalsofsomeeratthePostOffiumberofsutsiimeperiodtendstofollowsomePoissondistributioimetowaitbetweehisexpoialformat.

7.Gaussiandistributions

&importantuousdistributionistheonewehavealreadyhe　Gaussiandistribution.As　Figure　7illustrates，membersofthisfamilyaresymmetridasinglepeakandfallaidlytowardszero，whileuallyattainingthatvalue.Twoowhichmemberofthisfamilyanyexamplebelongs:onenumberpicksoutthepeak，theotherdescribesthespread–smallvaluesofthespreadleadtotallandnarrohslikeFigure　7a，largervaluesgiveshraphslike7yprobabilityforamemberofthisfamilybefoundbyusiwoorelateitto　Figure　7b，eakatzeroanditsmeasureofspreadstay.SuitabletableshavebeenwidelyavailablesincedeMoivrefirstproducedthem.

7.ued

Resolvinganissue

Youmayhavenoti.Providedthesetofsupposedoutesisfinite，oranunendinglistlike{1，2，3，...}，

thenevenifsomemembersofthislistturnouttohaveprobabilityzero，awhoseprobabilityiszerowillneverhappeha　uous　distribution，althougheachseparatepointhasprobabilityzero，ohem　willotheexperimentisperformed!Wegertake‘willnothappen’and‘probabilityzero’asmeahesame.

Toreatters，thinkofarbleatrandomfromaboxholdingamillioicalmarbles.Wewouldbeverysurprisedifwecuessedtheouteihegsoisonlyoneinamillion.Butwhichevermarblegets，wedonotthenexpresssurprise，eventhoughanoute，whoseprobabilitywasassmallasoneinamillion，hasoccurred.

&heboxbigger–abilliorillion–andthediheaebemadeasclosetozeroaswelike–butitdidhappen.gorandomonauousliverydifferentfromthis:foranypoint，itsceiszero，butohemwilloccur.

belowthat，iableexperimeheceofguessieisoneinsix，ethatexperimentsixtimesihtjustonce.Similarly，iftheillioakeamilliohtoheceofocebyafaothermillioimeweexpecttowaitfuessgetsmultipliedbyamilliohreallytinyprobabilitiesdooccur，butmoreandmorerarely.

Iftheprobabilityfallsallthewaydowntozero，ecttowaitlohanaime–itjusten!Itisrationaltoactasthoughaofprobabilityzero，thatisnamedinadvance，willnotoccur.

Meanvalues

Kributioaceexperiment，wecalyprobabilitywelike.Butsometimes，allthisdetailgetsintheway–we’tseethewoodforthetrees:soiaihedistribution.

Toillustrate，supposetheonlyoutespossibleare2，3，and7，withrespectiveprobabilities60%，10%，ahat，overahuionsofthisexperiment，thevalue2willoesixtytimes，3abouttentimes，and7theremainingthirtytimes.Thetotalofallthesevaluesis120+30+210=360，seoveralltheoesis360100=3.6.Thisathe　weightedsumofthevalues2，3，asbeingtheirprobabilities.

&ributionwehave，similarcalsleadteouteenumberofrepetitie’isalooseword，weprefertheterm　meaofthiscal.Theremaybeshortcuts:ifthevaluesare　uniformlydistributedoverse，themeanisjustmidwaybetweeremes;themeannumberofSuasequerialsultiplyingtherialsbytheceofSuccess.

Whenrollingafairdie，theceofgettingaFouris16.Soamong600throws，weshouldseearound100Fours:simplearithmetisaysthatthemeaweensuccessiveappearancesofaFouris6.Itisplainlynocethataceofsize16leadstoameangapof6.ThelengthofanygapisjustthetimetowaitfortheSuccess，soleasi，duringasequerials，

&imetowaitforaSuccessisthereciprocaloftheprobabilityofSuccess.

Withuousdistributions，theideaisthesame，buttheweightedsumisfoundbyusiiiqueknowiaussiandistributions，themeahepeakoccurs.Expoialdistributioimetowaitforara，whichocecharacteristicoverallfrequency:itshouldbenosurprisethatthemeaisjustthereciprocalofthatfrequency.

&erms‘expe’aedvalue’arealsousedinsteadof‘mean’and‘meaossingafaires，the‘expeumberofHeadsissix，aed’sthrowinganordinaryfairdieis3.5.Ofcourse，justbecausetheexpeumberofTailsoossis0.5，youdon’tactuallyexpecttogethalfaTail!TheEnglishlanguagehasmanyquirks.

Meansareveryfriendlyanimals:themeanofasumisalwaysthesumofthemeathedifferesariseiheLawehat，inthelongrun，meansdominate:ifyouspend￡1o，wherehalfthatsumgoesintotheprizefund，theheprizedistributionisstrueaurnis50pand，inthe（very）longrun，thatiswhatyouwillget.Variability

Itisalsousefultohaveasuctwaythevariabilityofadistribution.Wecouldcalculatethediffereweeneachvalueahenfindthe（prhted）averagevalueofthesediffere，asanytrialcalwillshow，thispathisfruitless:theivediffereablyexacelthepositiveones，alwaysgivingafinalanswerofzero.Butwhetheradiffereiveative，weositivenumberwhenwesquareit.Sowecouldusetheweightedaverageofthesesquaredvaluestoassessthevariability.Thisquantityiscalledthe　variahedistributioedhevariancewillbesmall;itwillbelargerwhenthereisareasonableceofgettingvalueswellawayfromthemean.

Whengiributions，withdataindollars，thesquareddataarein‘squaredollars’，whateverthatmightmean.Takingthesquarerootofthevariaurnsustiunits，givingwhatiscalledthe　staion.

Themeanandstaiveasfuliaiuresofaprobabilitydistribution.AndintheGaussiawonumberssuffidanyprobabilityatall!Asusefultoues，theoutewheributionisGaussiahiaionofthemeanabout68%ofthetime，withintwostaionsover95%ofthetime，andoimein400willitbemorethaaionsa>

ThesefiguresarethebasisfuidelinesofferedihowagreementweablyexpeSuccessprobability，aualfrequencyofSuccess:thekeyisthetralLimitTheorem，whichsaysthatquaariseasthesumenumberofrandompoedtofollowadistributioheGaussian.

In　Figure　7，shaussiayfuneansofthegraphsareat2，0，and2，whiletheirrespedarddeviationsare12，1，and2.

Butbewarhoughthemeanofasumisalwaysthesumofthemeans，thesameisrueofeitherthevariaaion.Ifthepohesumhappentobei–sayao’sprofitsoverseveedaysihenthevariahesumwillihesumoftheindividualvariaherwiseitcouldbehigherorlower.

Addingstaioherseldomleadsanywheresensible.

&reme-valuedistributions

Inseveralapplisofprobability，iresestorthesmallestenumberofrandomquantities.Forexample，thestrengthofathreadoracablerestsoiesoftheweakestfibre;flooddefeofthemaximumsurgethatmightbeexpectedoverthehuhesubjectofsurvivalanalysisexamifraofapopulatioeragiveremeeventsmayoccurrarely，butwhentheyhappen，thebeimportant.

&plausiblemodelassumesthereareirandomquantities，eachfolloarticulardistributioheclaimsmadeonaninsuranpanyieyear.Thepaedinhowbigisthelargesttotalclaimitexpecttoreceiveovertheyyears.Thereisausefulmathematicalresultthatgoesalongwaytthisquestioheclaimsvaryleyear，thereareonly　threepossibletypesofahemaximumclaimenumberofyears.Theyareknowreme-valuedistributions，withthespeamesoftheFréchet，theGumbel，andtheWeibulldistributions.Thereisasouiciplethatifthereissometheoremaboutmaxima，thereisadiaboutminima.Soiftheitemofiissomeminimumvalue，thesamepertains.

&olimitthepossibilitiestothesethreefamiliesofdistributionsisveryhelpful.Byestimatingthemeanandvarianextremevalue，aihemseemstofitthedatabest，seesofotherprobabilities–thecesofreallyextremeais–befound.