beta distribution

Key facts

Notation

Beta( α , β )

Parameters

α > 0 shape ( real ), β > 0 shape ( real )

Support

x ∈ [ 0 , 1 ] {\displaystyle x\in [0,1]\!} or x ∈ ( 0 , 1 ) {\displaystyle x\in (0,1)\!}

Cdf

I x ( α , β ) {\displaystyle I_{x}(\alpha ,\beta )\!} (the regularized incomplete beta function )

Mode

α − 1 α + β − 2 {\displaystyle {\frac {\alpha -1}{\alpha +\beta -2}}\!} for α , β > 1 Any value in the domain for α = β = 1 No mode if α <1 or β <1. Density diverges at 0 for α ≤ 1, and at 1 if β ≤ 1

Skewness

2 ( β − α ) α + β + 1 ( α + β + 2 ) α β {\displaystyle {\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}}

Excess kurtosis

6 [ ( α − β ) 2 ( α + β + 1 ) − α β ( α + β + 2 ) ] α β ( α + β + 2 ) ( α + β + 3 ) {\displaystyle {\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}

Entropy

ln ⁡ B ( α , β ) − ( α − 1 ) ψ ( α ) − ( β − 1 ) ψ ( β ) + ( α + β − 2 ) ψ ( α + β ) {\displaystyle {\begin{matrix}\ln \mathrm {B} (\alpha ,\beta )-(\alpha -1)\psi (\alpha )-(\beta -1)\psi (\beta )\\[0.5em]{}+(\alpha +\beta -2)\psi (\alpha +\beta )\end{matrix}}}

Mgf

1 + ∑ k = 1 ∞ ( ∏ r = 0 k − 1 α + r α + β + r ) t k k ! {\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}}

Cf

1 F 1 ( α ; α + β ; i t ) {\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!} (see Confluent hypergeometric function )

Method of moments

α = ( E [ X ] ( 1 − E [ X ] ) V [ X ] − 1 ) E [ X ] {\displaystyle \alpha =\left({\frac {E[X](1-E[X])}{V[X]}}-1\right)E[X]} , β = ( E [ X ] ( 1 − E [ X ] ) V [ X ] − 1 ) ( 1 − E [ X ] ) {\displaystyle \beta =\left({\frac {E[X](1-E[X])}{V[X]}}-1\right)(1-E[X])}

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Wikidata facts

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