# Distribusyong-t ni Student

Parameters Probability density function Cumulative distribution function ${\displaystyle \nu }$ > 0 degrees of freedom (real) x ∈ (−∞; +∞) ${\displaystyle \textstyle {\frac {\Gamma \left({\frac {\nu +1}{2}}\right)}{{\sqrt {\nu \pi }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\left(1+{\frac {x^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}\!}$ ${\displaystyle {\begin{matrix}{\frac {1}{2}}+x\Gamma \left({\frac {\nu +1}{2}}\right)\cdot \\[0.5em]{\frac {\,_{2}F_{1}\left({\frac {1}{2}},{\frac {\nu +1}{2}};{\frac {3}{2}};-{\frac {x^{2}}{\nu }}\right)}{{\sqrt {\pi \nu }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\end{matrix}}}$where 2F1 is the hypergeometric function 0 for ${\displaystyle \nu }$ > 1, otherwise undefined 0 0 ${\displaystyle \textstyle {\frac {\nu }{\nu -2}}}$ for ${\displaystyle \nu }$ > 2, ∞ for 1 < ${\displaystyle \nu }$ ≤ 2, otherwise undefined 0 for ${\displaystyle \nu }$ > 3, otherwise undefined ${\displaystyle \textstyle {\frac {6}{\nu -4}}}$ for ${\displaystyle \nu }$ > 4, ∞ for 2 < ${\displaystyle \nu }$ ≤ 4, otherwise undefined ${\displaystyle {\begin{matrix}{\frac {\nu +1}{2}}\left[\psi \left({\frac {1+\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)\right]\\[0.5em]+\log {\left[{\sqrt {\nu }}B\left({\frac {\nu }{2}},{\frac {1}{2}}\right)\right]}\end{matrix}}}$ undefined ${\displaystyle \textstyle {\frac {K_{\nu /2}\left({\sqrt {\nu }}|t|)({\sqrt {\nu }}|t|\right)^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}}}$ for ${\displaystyle \nu }$ > 0 ${\displaystyle K_{\nu }}$(x): Bessel function[1]