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SciPy

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Student t Distribution

Shape parameter ν>0. I(a,b,x) is the incomplete beta integral and I1(a,b,I(a,b,x))=x

f(x;ν)=Γ(ν+12)πνΓ(ν2)[1+x2ν]ν+12F(x;ν)={12I(ν2,12,νν+x2)x0112I(ν2,12,νν+x2)x0G(q;ν)={νI1(ν2,12,2q)νq12νI1(ν2,12,22q)νq12
mn=md=μ=0μ2=νν2ν>2γ1=0ν>3γ2=6ν4ν>4

As ν, this distribution approaches the standard normal distribution.

h[X]=14log(πcΓ2(c2)Γ2(c+12))(c+1)4[Ψ(c2)cZ(c)+πtan(πc2)+γ+2log2]

where

Z(c)=3F2(1,1,1+c2;32,2;1)=k=0k!k+1Γ(c2+1+k)Γ(c2+1)Γ(32)Γ(32+k)

Implementation: scipy.stats.t