Bernoulli Distribution: Definition and Examples - Statistics ...

A Bernoulli distribution is a discrete probability distribution for a Bernoulli trial — a random experiment that has only two outcomes ... ShareonProbabilityDistributions>BernoulliDistribution WhatisaBernoulliDistribution? ABernoullidistributionisadiscreteprobabilitydistributionforaBernoullitrial—arandomexperimentthathasonlytwooutcomes(usuallycalleda“Success”ora“Failure”).Forexample,theprobabilityofgettingaheads(a“success”)whileflippingacoinis0.5.Theprobabilityof“failure”is1–P(1minustheprobabilityofsuccess,whichalsoequals0.5foracointoss).Itisaspecialcaseofthebinomialdistributionforn=1.Inotherwords,itisabinomialdistributionwithasingletrial(e.g.asinglecointoss). WatchthevideoforthedefinitionandhowtofindthePDF,variance,expectedvalueandprobabilities: BernoulliDistributionProbability&PDFWatchthisvideoonYouTube. Can’tseethevideo?Clickhere. Theprobabilityofafailureislabeledonthex-axisas0andsuccessislabeledas1.InthefollowingBernoullidistribution,theprobabilityofsuccess(1)is0.4,andtheprobabilityoffailure(0)is0.6: Theprobabilitydensityfunction(pdf)forthisdistributionispx(1–p)1–x,whichcanalsobewrittenas: Theexpectedvalueforarandomvariable,X,foraBernoullidistributionis: E[X]=p. Forexample,ifp=.04,thenE[X]=0.04. ThevarianceofaBernoullirandomvariableis: Var[X]=p(1–p). WhatisaBernoulliTrial? ABernoullitrialisoneofthesimplestexperimentsyoucanconduct.It’sanexperimentwhereyoucanhaveoneoftwopossibleoutcomes.Forexample,“Yes”and“No”or“Heads”and“Tails.”Afewexamples: Cointosses:recordhowmanycoinslandheadsupandhowmanylandtailsup. Births:howmanyboysarebornandhowmanygirlsareborneachday. RollingDice:theprobabilityofarolloftwodieresultinginadoublesix. CointossingasagameofprobabilityandchancehasbeenaroundsinceRomantimes.Bernoullitrialsareusuallyphrasedintermsofsuccessandfailure.Successdoesn’tmeansuccessintheusualway—itjustreferstoanoutcomeyouwanttokeeptrackof.Forexample,youmightwanttofindouthowmanyboysareborneachday,soyoucallaboybirtha“success”andagirlbirtha“failure.”Inthedicerollingexample,adoublesixdierollwouldbeyour“success”andeverythingelserolledwouldbeconsidereda“failure.” Independence AnimportantpartofeveryBernoullitrialisthateachactionmustbeindependent.Thatmeanstheprobabilitiesmustremainthesamethroughoutthetrials;eacheventmustbecompletelyseparateandhavenothingtodowiththepreviousevent. Winningascratchofflotteryisanindependentevent.Youroddsofwinningononeticketarethesameaswinningonanyotherticket.Ontheotherhand,drawinglottonumbersisadependentevent.Lottonumberscomeoutofaball(thenumbersaren’treplaced)sotheprobabilityofsuccessivenumbersbeingpickeddependsuponhowmanyballsareleft;whenthere’sahundredballs,theprobabilityis1/100thatanynumberwillbepicked,butwhenthereareonlytenballsleft,theprobabilityshootsupto1/10.Whileit’spossibletofindthoseprobabilities,itisn’taBernoullitrialbecausetheevents(pickingthenumbers)areconnectedtoeachother. TheBernoulliprocessleadstoseveralprobabilitydistributions: Thebinomialdistribution, Thegeometricdistribution, Thenegativebinomialdistribution. TheBernoullidistributioniscloselyrelatedtotheBinomialdistribution.AslongaseachindividualBernoullitrialisindependent,thenthenumberofsuccessesinaseriesofBernoullitrailshasaBinomialDistribution.TheBernoullidistributioncanalsobedefinedastheBinomialdistributionwithn=1. UseoftheBernoulliDistributioninEpidemiology Inexperimentsandclinicaltrials,theBernoullidistributionissometimesusedtomodelasingleindividualexperiencinganeventlikedeath,adisease,ordiseaseexposure.Themodelisanexcellentindicatoroftheprobabilityapersonhastheeventinquestion. 1=“event”(P=p) 0=“nonevent”(P=1–p) Bernoullidistributionsareusedinlogisticregressiontomodeldiseaseoccurrence. References Evans,M.;Hastings,N.;andPeacock,B.“BernoulliDistribution.”Ch.4inStatisticalDistributions,3rded.NewYork:Wiley,pp.31-33,2000. WSU.RetrievedFeb15,2016from:www.stat.washington.edu/peter/341/Hypergeometric%20and%20binomial.pdf CITETHISAS:StephanieGlen."BernoulliDistribution:DefinitionandExamples"FromStatisticsHowTo.com:ElementaryStatisticsfortherestofus!https://www.statisticshowto.com/bernoulli-distribution/ --------------------------------------------------------------------------- Needhelpwithahomeworkortestquestion?WithCheggStudy,youcangetstep-by-stepsolutionstoyourquestionsfromanexpertinthefield.Yourfirst30minuteswithaCheggtutorisfree! Comments?Needtopostacorrection?PleasepostacommentonourFacebookpage. StatisticsHowToFacebookPage ByStephanie| August16,2021| StatisticsHowTo| ←Factorial:SimpleDefinition,Examples&Distribution BoxCoxTransformation:Definition,Examples→ Search NEEDHELPwithahomeworkproblem?CLICKHERE!Needhelpwithahomeworkortestquestion?WithCheggStudy,youcangetstep-by-stepsolutionstoyourquestionsfromanexpertinthefield.Yourfirst30minuteswithaCheggtutorisfree! 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