.. _continuous-ncx2:

Noncentral chi-squared Distribution
===================================

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.


References
----------

-  "Noncentral chi-squared distribution", Wikipedia
   https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

Implementation: `scipy.stats.ncx2`