Notation Probability density function Cumulative distribution function ${\displaystyle \chi ^{2}(k)\!}$ or ${\displaystyle \chi _{k}^{2}\!}$ ${\displaystyle k\in \mathbb {N} ~~}$ (known as "degrees of freedom") x ∈ [0, +∞) ${\displaystyle {\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\,}$ ${\displaystyle {\frac {1}{\Gamma \left({\frac {k}{2}}\right)}}\;\gamma \left({\frac {k}{2}},\,{\frac {x}{2}}\right)}$ k ${\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}}$ max{ k − 2, 0 } 2k ${\displaystyle \scriptstyle {\sqrt {8/k}}\,}$ 12 / k ${\displaystyle {\frac {k}{2}}\!+\!\ln(2\Gamma (k/2))\!+\!(1\!-\!k/2)\psi (k/2)}$ (1 − 2 t)−k/2   for  t  < ½ (1 − 2 i t)−k/2      [1]