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Student t Distribution¶
Shape parameter ν>0. I(a,b,x) is the incomplete beta integral and I−1(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)x≤01−12I(ν2,12,νν+x2)x≥0G(q;ν)={−√νI−1(ν2,12,2q)−νq≤12√νI−1(ν2,12,2−2q)−νq≥12
mn=md=μ=0μ2=νν−2ν>2γ1=0ν>3γ2=6ν−4ν>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