Processing math: 100%

Beta Prime Distribution

Defined over 0<x<. α,β>0. (Note the CDF evaluation uses Eq. 3.194.1 on pg. 313 of Gradshteyn & Ryzhik (sixth edition).

f(x;α,β)=Γ(α+β)Γ(α)Γ(β)xα1(1+x)αβF(x;α,β)=Γ(α+β)αΓ(α)Γ(β)xα2F1(α+β,α;1+α;x)G(q;α,β)=F1(x;α,β)
μn={Γ(n+α)Γ(βn)Γ(α)Γ(β)=(α)n(βn)nβ>notherwise

Therefore,

μ=αβ1β>1μ2=α(α+1)(β2)(β1)α2(β1)2β>2γ1=α(α+1)(α+2)(β3)(β2)(β1)3μμ2μ3μ3/22β>3γ2=μ4μ223μ4=α(α+1)(α+2)(α+3)(β4)(β3)(β2)(β1)4μμ36μ2μ2μ4β>4

Implementation: scipy.stats.betaprime