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SciPy

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Inverted Gamma Distribution

Special case of the generalized Gamma distribution with c=1 and a>0 , x>0

f(x;a)=xa1Γ(a)exp(1x)F(x;a)=Γ(a,1x)Γ(a)G(q;a)={Γ1[a,Γ(a)q]}1
μn=Γ(an)Γ(a)a>n
μ=1a1a>1μ2=1(a2)(a1)μ2a>2γ1=1(a3)(a2)(a1)3μμ2μ3μ3/22γ2=1(a4)(a3)(a2)(a1)4μμ36μ2μ2μ4μ223
md=1a+1
h[X]=a(a+1)Ψ(a)+logΓ(a).

Implementation: scipy.stats.invgamma