一般化双曲型分布
自由度 ν の(対称な)スチューデントのt分布となる。
g h ( x ; λ = − ν 2 , α = 0 , β = 0 , δ = ν , μ ) = Γ ( ν + 1 2 ) π δ Γ ( ν 2 ) [ 1 + ( x − μ ) 2 δ 2 ] − ν + 1 2 = Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) ( 1 + ( x − μ ) 2 ν ) − ν + 1 2 {\displaystyle {\begin{aligned}gh(x;&\lambda ={\frac {-\nu }{2}},\alpha =0,\beta =0,\delta ={\sqrt {\nu }},\mu )\\&={\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{{\sqrt {\pi }}\delta \Gamma \left({\frac {\nu }{2}}\right)}}\left[1+{\frac {(x-\mu )^{2}}{\delta ^{2}}}\right]^{-{\frac {\nu +1}{2}}}\\&={\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{{\sqrt {\pi \nu }}\Gamma \left({\frac {\nu }{2}}\right)}}\left(1+{\frac {(x-\mu )^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}\end{aligned}}}