The noncentral chi-square distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The NonCentralChiSquare variate with noncentrality parameter delta=0 and degrees of freedom nu is equivalent to the ChiSquare variate with degrees of freedom nu.

Note that the NonCentralChiSquare command is inert and should be used in combination with the RandomVariable command.

The Quantile and CDF functions applied to a noncentral chi-square distribution use a sequence of iterations in order to converge on the desired output point.  The maximum number of iterations to perform is equal to 100 by default, but this value can be changed by setting the environment variable _EnvStatisticsIterations to the desired number of iterations.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{NonCentralChiSquare}\left(\mathrm{\nu }\,\mathrm{\delta }\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\left\{\begin{array}{cc}0& u<0\\ \frac{{\&ExponentialE;}^{-\frac{u}{2}-\frac{\mathrm{\delta }}{2}}{u}^{\frac{\mathrm{\nu }}{2}-1}\mathrm{hypergeom}\left(\left[\right]\&comma;\left[\frac{\mathrm{\nu }}{2}\right]\&comma;\frac{\mathrm{\delta }u}{4}\right)}{\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right)2^{\frac{\mathrm{\nu }}{2}}}& \mathrm{otherwise}\end{array}\right.$

$\mathrm{PDF}\left(X\&comma;\frac{1}{2}\right)$

$\frac{{\&ExponentialE;}^{-\frac{1}{4}-\frac{\mathrm{\delta }}{2}}{\left(\frac{1}{2}\right)}^{\frac{\mathrm{\nu }}{2}-1}\mathrm{hypergeom}\left(\left[\right]\&comma;\left[\frac{\mathrm{\nu }}{2}\right]\&comma;\frac{\mathrm{\delta }}{8}\right)}{\mathrm{\Gamma }\left(\frac{\mathrm{\nu }}{2}\right)2^{\frac{\mathrm{\nu }}{2}}}$

$\mathrm{Mean}\left(X\right)$

$\mathrm{\nu }+\mathrm{\delta }$

$\mathrm{Variance}\left(X\right)$

$2\mathrm{\nu }+4\mathrm{\delta }$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.