- Notation
- χ 2 ( k ) {\displaystyle \chi ^{2}(k)\;} or χ k 2 {\displaystyle \chi _{k}^{2}\!}
- Parameters
- k ∈ N ∗ {\displaystyle k\in \mathbb {N} ^{*}~~} (known as "degrees of freedom")
- Support
- x ∈ ( 0 , + ∞ ) {\displaystyle x\in (0,+\infty )\;}
- Pdf
- 1 2 k / 2 Γ ( k / 2 ) x ( k / 2 ) − 1 e − x / 2 {\displaystyle {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{(k/2)-1}e^{-x/2}\;}
- Cdf
- 1 Γ ( k / 2 ) γ ( k 2 , x 2 ) {\displaystyle {\frac {1}{\Gamma (k/2)}}\;\gamma {\left({\frac {k}{2}},\,{\frac {x}{2}}\right)}\;}
- Mean
- k {\displaystyle k}
- Median
- ≈ k ( 1 − 2 9 k ) 3 {\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;}
- Mode
- max ( k − 2 , 0 ) {\displaystyle \max(k-2,0)\;}
- Variance
- 2 k {\displaystyle 2k\;}
- Skewness
- 8 / k {\textstyle {\sqrt {8/k}}\,}
- Excess kurtosis
- 12 k {\displaystyle {\frac {12}{k}}}
- Entropy
- k 2 + log ( 2 Γ ( k 2 ) ) + ( 1 − k 2 ) ψ ( k 2 ) {\displaystyle {\begin{aligned}{\frac {k}{2}}&+\log \left(2\Gamma {\left({\frac {k}{2}}\right)}\right)\\&\!+\left(1-{\frac {k}{2}}\right)\psi {\left({\frac {k}{2}}\right)}\end{aligned}}}
- Mgf
- ( 1 − 2 t ) − k / 2 {\displaystyle (1-2t)^{-k/2}} for t < 1 2 {\displaystyle t<{\tfrac {1}{2}}\;}
- Cf
- ( 1 − 2 i t ) − k / 2 {\displaystyle (1-2it)^{-k/2}}
- Pgf
- ( 1 − 2 ln t ) − k / 2 {\displaystyle (1-2\ln t)^{-k/2}} for 0 < t < e {\displaystyle 0<t<{\sqrt {e}}\;}