Noncentral chi distribution

Noncentral chi

Parameters	${\displaystyle k>0\,}$ degrees of freedom ${\displaystyle \lambda >0\,}$
Support	${\displaystyle x\in [0;+\infty )\,}$
PDF	${\displaystyle {\frac {e^{-(x^{2}+\lambda ^{2})/2}x^{k}\lambda }{(\lambda x)^{k/2}}}I_{k/2-1}(\lambda x)}$
CDF	${\displaystyle 1-Q_{\frac {k}{2}}\left(\lambda ,x\right)}$ with Marcum Q-function ${\displaystyle Q_{M}(a,b)}$
Mean	${\displaystyle {\sqrt {\frac {\pi }{2}}}L_{1/2}^{(k/2-1)}\left({\frac {-\lambda ^{2}}{2}}\right)\,}$
Variance	${\displaystyle k+\lambda ^{2}-\mu ^{2}\,}$

${\displaystyle k>0\,}$ degrees of freedom

In probability theory and statistics, the noncentral chi distribution is a generalization of the chi distribution. If ${\displaystyle X_{i}}$ are k independent, normally distributed random variables with means ${\displaystyle \mu _{i}}$ and variances ${\displaystyle \sigma _{i}^{2}}$, then the statistic

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