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SciPy

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Noncentral t Distribution

The distribution of the ratio

U+λχν/ν

where U and χν are independent and distributed as a standard normal and chi with ν degrees of freedom. Note λ>0 and ν>0 .

f(x;λ,ν)=νν/2Γ(ν+1)2νeλ2/2(ν+x2)ν/2Γ(ν/2)×{2λx1F1(ν2+1;32;λ2x22(ν+x2))(ν+x2)Γ(ν+12)1F1(ν+12;12;λ2x22(ν+x2))ν+x2Γ(ν2+1)}=Γ(ν+1)2(ν1)/2πνΓ(ν/2)exp[νλ2ν+x2]×(νν+x2)(ν1)/2Hhν(λxν+x2)F(x;λ,ν)=

Implementation: scipy.stats.nct