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Beta distribution

Probability density function:

f(x)=xα1(1x)β1B(α,β) with α and β two shape parameters and B beta function.

Cumulative distribution function:

F(x)=x0yα1(1y)β1dyB(α,β)=B(x;α,β) with B(x;α,β) incomplete beta function.

Log-likelihood function:

L(α,β;X)=i[(α1)ln(x)+(β1)ln(1x)lnB(α,β)]

Score function vector:

V(μ,σ;X)=(LαLβ)=i(ψ(0)(α+β)ψ(0)(α)+ln(x)ψ(0)(α+β)ψ(0)(β)+ln(x)) with ψ(0) being log-gamma function.

Observed information matrix:

J(μ,σ;X)=(ψ(1)(α)ψ(1)(α+β)ψ(1)(α+β)ψ(1)(α+β)ψ(1)(β)ψ(1)(α+β)) with ψ(1) being digamma function.