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BetadistributionTheBetadistributionisacontinuousprobabilitydistributionhavingtwoparameters.Oneofitsmostcommonusesistomodelone'suncertaintyabouttheprobabilityofsuccessofanexperiment.

Supposeaprobabilisticexperimentcanhaveonlytwooutcomes,eithersuccess,withprobability ,orfailure,withprobability .Supposealsothat isunknownandallitspossiblevaluesaredeemedequallylikely.Thisuncertaintycanbedescribedbyassigningto auniformdistributionontheinterval .Thisisappropriatebecause ,beingaprobability,cantakeonlyvaluesbetween and furthermore,theuniformdistributionassignsequalprobabilitydensitytoallpointsintheinterval,whichreflectsthefactthatnopossiblevalueof is,apriori,deemedmorelikelythanalltheothers.Now,supposethatweperform independentrepetitionsoftheexperimentandweobserve successesand failures.Afterperformingtheexperiments,wenaturallywanttoknowhowweshouldrevisethedistributioninitiallyassignedto ,inordertoproperlytakeintoaccounttheinformationprovidedbytheobservedoutcomes.Inotherwords,wewanttocalculatetheconditionaldistributionof ,conditionalonthenumberofsuccessesandfailureswehaveobserved.TheresultofthiscalculationisaBetadistribution.Inparticular,theconditionaldistributionof ,conditionalonhavingobserved successesoutof trials,isaBetadistributionwithparameters and .

DefinitionTheBetadistributionischaracterizedasfollows.

Definition Let beanabsolutelycontinuousrandomvariable.Letitssupportbetheunitinterval:

Let .Wesaythat hasaBetadistributionwithshapeparameters and ifitsprobabilitydensityfunctionis

where istheBetafunction.

ArandomvariablehavingaBetadistributionisalsocalledaBetarandomvariable.

Thefollowingisaproofthat isalegitimateprobabilitydensityfunction.

Proof

Nonnegativitydescendsfromthefactsthat isnonnegativewhen and ,andthat isstrictlypositive(itisaratioofGammafunctions,whicharestrictlypositivewhentheirargumentsarestrictlypositiveseethelectureentitledGammafunction).Thattheintegralof over equals isprovedasfollows:

wherewehaveusedtheintegralrepresentation

aproofofwhichcanbefoundinthelectureentitledBetafunction.

ExpectedvalueTheexpectedvalueofaBetarandomvariable is

Proof

Itcanbederivedasfollows:

VarianceThevarianceofaBetarandomvariable is

Proof

Itcanbederivedthankstotheusualvarianceformula( ):

HighermomentsThe thmomentofaBetarandomvariable is

Proof

Bythedefinitionofmoment,wehave:

whereinstep wehaveusedrecursivelythefactthat .

MomentgeneratingfunctionThemomentgeneratingfunctionofaBetarandomvariable isdefinedforany anditis

Proof

Byusingthedefinitionofmomentgeneratingfunction,weobtain

Notethatthemomentgeneratingfunctionexistsandiswelldefinedforany becausetheintegral

isguaranteedtoexistandbefinite,sincetheintegrand

iscontinuousin overtheboundedinterval .

Theaboveformulaforthemomentgeneratingfunctionmightseemimpracticaltocompute,becauseitinvolvesaninfinitesumaswellasproductswhosenumberoftermsincreaseindefinitely.However,thefunction

isafunction,calledConfluenthypergeometricfunctionofthefirstkind,thathasbeenextensivelystudiedinmanybranchesofmathematics.Itspropertiesarewellknownandefficientalgorithmsforitscomputationareavailableinmostsoftwarepackagesforscientificcomputation.

CharacteristicfunctionThecharacteristicfunctionofaBetarandomvariable is

Proof

Thederivationofthecharacteristicfunctionisalmostidenticaltothederivationofthemomentgeneratingfunction(justreplace with inthatproof).

Commentsmadeaboutthemomentgeneratingfunction,includingthoseaboutthecomputationoftheConfluenthypergeometricfunction,applyalsotothecharacteristicfunction,whichisidenticaltothemgfexceptforthefactthatisreplacedwith .

DistributionfunctionThedistributionfunctionofaBetarandomvariable is

wherethefunction

iscalledincompleteBetafunctionandisusuallycomputedbymeansofspecializedcomputeralgorithms.

Proof

For , ,because cannotbesmallerthan .For , ,because isalwayssmallerthanorequalto .For ,

MoredetailsInthefollowingsubsectionsyoucanfindmoredetailsabouttheBetadistribution.

Relationtotheuniformdistribution

ThefollowingpropositionstatestherelationbetweentheBetaandtheuniformdistributions.

Proposition ABetadistributionwithparameters and isauniformdistributionontheinterval .

Proof

When and ,wehavethat

Therefore,theprobabilitydensityfunctionofaBetadistributionwithparameters and canbewrittenas

Butthelatteristheprobabilitydensityfunctionofauniformdistributionontheinterval .

Relationtothebinomialdistribution

ThefollowingpropositionstatestherelationbetweentheBetaandthebinomialdistributions.

Proposition Suppose isarandomvariablehavingaBetadistributionwithparameters and .Let beanotherrandomvariablesuchthatitsdistributionconditionalon isabinomialdistributionwithparametersand .Then,theconditionaldistributionof given isaBetadistributionwithparameters and

.

Proof

Bycombiningthispropositionandthepreviousone,weobtainthefollowingcorollary.

Proposition Suppose isarandomvariablehavingauniformdistribution.Let beanotherrandomvariablesuchthatitsdistributionconditionalon isabinomialdistributionwithparameters and .Then,theconditionaldistributionof given isaBetadistributionwithparameters and .

ThispropositionconstitutesaformalstatementofwhatwesaidintheintroductionofthislectureinordertomotivatetheBetadistribution.Rememberthatthenumberofsuccessesobtainedin independentrepetitionsofarandomexperimenthavingprobabilityofsuccess isabinomialrandomvariablewithparameters and .Accordingtothepropositionabove,whentheprobabilityofsuccess isaprioriunknownandallpossiblevaluesof aredeemedequallylikely(theyhaveauniformdistribution),observingtheoutcomeofthe experimentsleadsustorevisethedistributionassignedto ,andtheresultofthisrevisionisaBetadistribution.

SolvedexercisesBelowyoucanfindsomeexerciseswithexplainedsolutions:

1. Exerciseset1

FeaturedpagesMultivariatenormaldistributionHypothesistestingDeltamethodBernoullidistributionBayesruleIndependentevents

ExploreCentralLimitTheoremConvergenceindistributionMultinomialdistributionFdistribution

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