Noncentral chi-squared distribution

Noncentral chi-squared

Probability density function
Cumulative distribution function
Parameters	${\displaystyle k>0\,}$ degrees of freedom ${\displaystyle \lambda >0\,}$ non-centrality parameter
Support	${\displaystyle x\in [0;+\infty )\,}$
PDF	${\displaystyle {\frac {1}{2}}e^{-(x+\lambda )/2}\left({\frac {x}{\lambda }}\right)^{k/4-1/2}I_{k/2-1}({\sqrt {\lambda x}})}$
CDF	${\displaystyle 1-Q_{\frac {k}{2}}\left({\sqrt {\lambda }},{\sqrt {x}}\right)}$ with Marcum Q-function ${\displaystyle Q_{M}(a,b)}$
Mean	${\displaystyle k+\lambda \,}$
Variance	${\displaystyle 2(k+2\lambda )\,}$
Skewness	${\displaystyle {\frac {2^{3/2}(k+3\lambda )}{(k+2\lambda )^{3/2}}}}$
Ex. kurtosis	${\displaystyle {\frac {12(k+4\lambda )}{(k+2\lambda )^{2}}}}$
MGF	${\displaystyle {\frac {\exp \left({\frac {\lambda t}{1-2t}}\right)}{(1-2t)^{k/2}}}{\text{ for }}2t<1}$
CF	${\displaystyle {\frac {\exp \left({\frac {i\lambda t}{1-2it}}\right)}{(1-2it)^{k/2}}}}$

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

In probability theory and statistics, the noncentral chi-squared or noncentral ${\displaystyle \chi ^{2}}$ distribution is a generalization of the chi-squared distribution. This distribution often arises in the power analysis of statistical tests in which the null distribution is (perhaps asymptotically) a chi-squared distribution; important examples of such tests are the likelihood ratio tests.

Let (${\displaystyle X_{1}}$, ${\displaystyle X_{2},\ldots ,}$${\displaystyle X_{i},\ldots ,}$ ${\displaystyle X_{k}}$) be k independent, normally distributed random variables with means ${\displaystyle \mu _{i}}$ and unit variances. Then the random variable