Beta Distribution

Author

Eric W. Weisstein
September 29, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/BetaDistribution.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BetaDistribution.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Stats`

Plots

range[PDF]:={{0,1},{0,2.5}}
range[CDF]:={{0,1},{0,1}}

Show[
GraphicsArray[Function[f,Plot[Evaluate[f[BetaDistribution[Sequence@@#],x]&/@({1,#}&/@Range[.25,3,.25])],{x,0,1},AxesLabel->TraditionalForm/@{x,f[x]},Ticks->None,
PlotRange->range[f],
PlotStyle->Hue/@Range[0,1,.1],DisplayFunction->Identity,PlotRange->All]]/@{PDF,CDF}]
]

Stats

Stats[BetaDistribution[α,β],x]//StatsForm//TraditionalForm

Distribution Function

Simplify[FunctionExpand[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β]))]]

FullSimplify[Integrate[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β])),{x,0,x}],α>0]

Characteristic Function

CharacteristicFunction[BetaDistribution[α,β],t]

v4.2

Integrate[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β]))Exp[I t x],{x,0,1},Assumptions->{α>0,β>0,t>0}]

FullSimplify[%,α>0&&β>0&&t>0]

With[{α=1.1,β=2.2},
Plot[(Hypergeometric1F1[α,α+β,I t]-(2^(-1-α-β)) π Gamma[α+β] (4 HypergeometricPFQRegularized[{(α/2),((1+α)/2)},{(1/2),((α+β)/2),(1/2) (1+α+β)},-((t^2)/4)]+I t α HypergeometricPFQRegularized[{((1+α)/2),((2+α)/2)},{(3/2),(1/2) (1+α+β),(1/2) (2+α+β)},-((t^2)/4)]))//Re,{t,0,2},PlotStyle->Red]
]

v5.0

Integrate[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β]))Exp[I t x],{x,0,1}]

Integrate[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β]))Exp[I t x],{x,0,1},Assumptions->{α>0,β>0}]

FullSimplify[%]

Normalization

FullSimplify[Integrate[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β])),{x,0,1},Assumptions->α>0&&β>0]]

Mean

FullSimplify[Integrate[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β]))x,{x,0,1},Assumptions->α>0&&β>0]]

Raw Moments

FullSimplify[Integrate[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β]))x^r /. μ->(α/(α+β)),{x,0,1},GenerateConditions->False]]

FullSimplify[Table[Gamma[α+β]Gamma[α+r]/Gamma[α+β+r]/Gamma[α],{r,0,4}]]

Central Moments

FullSimplify[Integrate[(((((1-x))^(-1+β)) (x^(-1+α)))/(Beta[α,β]))(x-μ)^r /. μ->(α/(α+β)),{x,0,1},GenerateConditions->False]]

FullSimplify[Table[(((-(α/(α+β))))^r) Hypergeometric2F1[-r,α,α+β,((α+β)/α)],{r,0,4}]]

PDF[BetaDistribution[1/2,1/2],x]