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1、DiagramofdistributionrelationshipsProbabilitydistributionshaveasurprisingnumberinter-connections.Adashedlineinthechartbelowindicatesanapproximate(limit)relationshipbetweentwodistributionfamilies.Asolidlineindicatesanexactrelationship:specialcase,sum,ortransformation.Clickonadistributionforthep
2、arameterizationofthatdistribution.Clickonanarrowfordetailsontherelationshiprepresentedbythearrow.Otherdiagramsonthissite:ThechartaboveisadaptedfromthechartoriginallypublishedbyLawrenceLeemisin1986(RelationshipsAmongCommonUnivariateDistributions,AmericanStatistician40:143-146.)Leemispublishedal
3、argerchartin2008whichisavailableonline.ParameterizationsThepreciserelationshipsbetweendistributionsdependonparameterization.TherelationshipsdetailedbelowdependonthefollowingparameterizationsforthePDFs.LetC(n,k)denotethebinomialcoefficient(n,k)andB(a,b)=Γ(a)Γ(b)/Γ(a+b).Geometric:f(x)=p(1-p)xfor
4、non-negativeintegersx.Discreteuniform:f(x)=1/nforx=1,2,...,n.Negativebinomial:f(x)=C(r+x-1,x)pr(1-p)xfornon-negativeintegersx.Seenotesonthenegativebinomialdistribution.Betabinomial:f(x)=C(n,x)B(α+x,n+β-x)/B(α,β)forx=0,1,...,n.Hypergeometric:f(x)=C(M,x)C(N-M,K-x)/C(N,K)forx=0,1,...,N.Poisson:f(
5、x)=exp(-λ)λx/x!fornon-negativeintegersx.Theparameterλisboththemeanandthevariance.Binomial:f(x)=C(n,x)px(1-p)n-xforx=0,1,...,n.Bernoulli:f(x)=px(1-p)1-xwherex=0or1.Lognormal:f(x)=(2πσ2)-1/2exp(-(log(x)-μ)2/2σ2)/xforpositivex.Notethatμandσ2arenotthemeanandvarianceofthedistribution.Normal:f(x)=(2
6、πσ2)-1/2exp(-½((x-μ)/σ)2)forallx.Beta:f(x)=Γ(α+β)xα-1(1-x)β-1/(Γ(α)Γ(β))for0≤x≤1.Standardnormal:f(x)=(2π)-1/2exp(-x2/2)forallx.Chi-squared:f(x)=x-ν/2-1exp(-x/2)/Γ(ν/2)2ν/2forpositivex.Theparameterνiscalledthedegreesoffreedom.Gamma:f(x)=β-αxα-1exp(-x/β)/Γ(α)forpositivex.Theparameterαiscalledthe
7、shapeandβisthescale.Uniform:f(x)=1for0≤x≤1.Cauchy:f(x)=σ/(π((x-μ)2+σ2))forallx.Notethatμandσarelocationandscaleparameters.TheCauchydistributionhasnomeanorvariance.SnedecorF:f(x)isproportionaltox(ν1-2)/2/(1+(ν1/ν2)x)(ν1+ν2)/2forpositivex
