# Statistics/Distributions/Student-t

Parameters Probability density function Cumulative distribution function ν > 0 degrees of freedom (real) x ∈ (−∞; +∞) $\textstyle\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!$ $\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right) \cdot\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2}; -\frac{x^2}{\nu} \right)} {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)} \end{matrix}$ where 2F1 is the hypergeometric function 0 for ν > 1, otherwise undefined 0 0 $\textstyle\frac{\nu}{\nu-2}$ for ν > 2, ∞ for 1 < ν ≤ 2, otherwise undefined 0 for ν > 3, otherwise undefined $\textstyle\frac{6}{\nu-4}$ for ν > 4, ∞ for 2 < ν ≤ 4, otherwise undefined ... undefined $\textstyle\frac{K_{\nu/2} \left(\sqrt{\nu}|t|)(\sqrt{\nu}|t| \right)^{\nu/2}}{\Gamma(\nu/2)2^{\nu/2-1}}$ for ν > 0 $K_{\nu}$(x): Bessel function[1]
$t = \frac{\mbox{Z}}{ \sqrt{ \chi_{r}^2 /r }}$
where $Z = \frac{X - \bar{X}}{\sigma}$ and $\chi_{r}^2 = \chi_{r_1}^2 + ... + \chi_{r_n}^2$.