doc/source/tutorial/stats/continuous_ncx2.rst
.. _continuous-ncx2:
The distribution of :math:\sum_{i=1}^{\nu}\left(Z_{i}+\delta_{i}\right)^{2}
where :math:Z_{i} are independent standard normal variables and
:math:\delta_{i} are constants.
:math:\lambda=\sum_{i=1}^{\nu}\delta_{i}^{2}>0.
(In communications it is called the Marcum-Q function).
It can be thought of as a Generalized Rayleigh-Rice distribution.
The two shape parameters are :math:\nu, a positive integer, and :math:\lambda,
a positive real number. The support is :math:x\geq0.
.. math:: :nowrap:
\begin{eqnarray*} f\left(x;\nu,\lambda\right) & = & e^{-\left(\lambda+x\right)/2}\frac{1}{2}\left(\frac{x}{\lambda}\right)^{\left(\nu-2\right)/4}I_{\left(\nu-2\right)/2}\left(\sqrt{\lambda x}\right)\\
F\left(x;\nu,\lambda\right) & = & \sum_{j=0}^{\infty}\left\{ \frac{\left(\lambda/2\right)^{j}}{j!}e^{-\lambda/2}\right\} \mathrm{Pr}\left[\chi_{\nu+2j}^{2}\leq x\right]\\
G\left(q;\nu,\lambda\right) & = & F^{-1}\left(q;\nu,\lambda\right)\\
\mu & = & \nu+\lambda\\
\mu_{2} & = & 2\left(\nu+2\lambda\right)\\
\gamma_{1} & = & \frac{\sqrt{8}\left(\nu+3\lambda\right)}{\left(\nu+2\lambda\right)^{3/2}}\\
\gamma_{2} & = & \frac{12\left(\nu+4\lambda\right)}{\left(\nu+2\lambda\right)^{2}}\end{eqnarray*}
where :math:I_{\nu }(y) is a modified Bessel function of the first kind.
Implementation: scipy.stats.ncx2