Beta Distribution
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Search Go Statlect The Digital Textbook Index > Probability distributions Beta distribution The Beta distribution is a continuous probability distribution having two parameters. One of its most common uses is to model one's uncertainty about the probability of success of an experiment. Suppose a probabilistic experiment can have only two outcomes, either success, with probability , or failure, with probability . Suppose also that is unknown and all its possible values are deemed equally likely. This uncertainty can be described by assigning to a uniform distribution on the interval . This is appropriate because , being a probability, can take only values between and ; furthermore, the uniform distribution assigns equal probability density to all points in the interval, which reflects the fact that no possible value of is, a priori, deemed more likely than all the others. Now, suppose that we perform independent repetitions of the experiment and we observe successes and failures. After performing the experiments, we naturally want to know how we should revise the distribution initially assigned to , in order to properly take into account the information provided by the observed outcomes. In other words, we want to calculate the conditional distribution of , conditional on the number of successes and failures we have observed. The result of this calculation is a Beta distribution. In particular, the conditional distribution of , conditional on having observed successes out of trials, is a Beta distribution with parameters and . Definition The Beta distribution is characterized as follows. Definition Let be an absolutely continuous random variable. Let its support be the unit interval: Let . We say that has a Beta distribution with shape parameters and if its probability density function is where is the Beta function. A random variable having a Beta distribution is also called a Beta random variable. The following is a proof that is a legitimate probability density function. Proof Nonnegativity descends from the facts that is nonnegative when and , and that is strictly positive (it is a ratio of Gamma functions, which are strictly positive when their arguments are strictly positive see the lecture entitled Gamma function). That the integral of over equals is proved as follows:
description
Beta distribution
Transcript of Beta Distribution
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Search GoStatlect TheDigitalTextbookIndex>Probabilitydistributions
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:
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wherewehaveusedtheintegralrepresentation
aproofofwhichcanbefoundinthelectureentitledBetafunction.
ExpectedvalueTheexpectedvalueofaBetarandomvariable is
Proof
Itcanbederivedasfollows:
VarianceThevarianceofaBetarandomvariable is
Proof
Itcanbederivedthankstotheusualvarianceformula( ):
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HighermomentsThe thmomentofaBetarandomvariable is
Proof
Bythedefinitionofmoment,wehave:
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whereinstep wehaveusedrecursivelythefactthat .
MomentgeneratingfunctionThemomentgeneratingfunctionofaBetarandomvariable isdefinedforany anditis
Proof
Byusingthedefinitionofmomentgeneratingfunction,weobtain
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Notethatthemomentgeneratingfunctionexistsandiswelldefinedforany becausetheintegral
isguaranteedtoexistandbefinite,sincetheintegrand
iscontinuousin overtheboundedinterval .
Theaboveformulaforthemomentgeneratingfunctionmightseemimpracticaltocompute,becauseitinvolvesaninfinitesumaswellasproductswhosenumberoftermsincreaseindefinitely.However,thefunction
isafunction,calledConfluenthypergeometricfunctionofthefirstkind,thathasbeenextensivelystudiedinmanybranchesofmathematics.Itspropertiesarewellknownandefficientalgorithmsforitscomputationareavailableinmostsoftwarepackagesforscientificcomputation.
CharacteristicfunctionThecharacteristicfunctionofaBetarandomvariable is
Proof
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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
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Therefore,theprobabilitydensityfunctionofaBetadistributionwithparameters and canbewrittenas
Butthelatteristheprobabilitydensityfunctionofauniformdistributionontheinterval .
Relationtothebinomialdistribution
ThefollowingpropositionstatestherelationbetweentheBetaandthebinomialdistributions.
Proposition Suppose isarandomvariablehavingaBetadistributionwithparameters and .Let beanotherrandomvariablesuchthatitsdistributionconditionalon isabinomialdistributionwithparametersand .Then,theconditionaldistributionof given isaBetadistributionwithparameters and
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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
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