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.. _continuous-ncf:
Noncentral F Distribution
=========================
The distribution of :math:`\left(X_{1}/X_{2}\right)\left(\nu_{2}/\nu_{1}\right)`
if :math:`X_{1}` is non-central chi-squared with :math:`\nu_{1}` degrees of
freedom and parameter :math:`\lambda`, and :math:`X_{2}` is chi-squared with
:math:`\nu_{2}` degrees of freedom.
There are 3 shape parameters: the degrees of freedom :math:`\nu_{1}>0` and
:math:`\nu_{2}>0`; and :math:`\lambda\geq 0`.
.. math::
:nowrap:
\begin{eqnarray*}
f\left(x;\lambda,\nu_{1},\nu_{2}\right)
& = &
\exp\left[\frac{\lambda}{2} +
\frac{\left(\lambda\nu_{1}x\right)}
{2\left(\nu_{1}x+\nu_{2}\right)}
\right]
\nu_{1}^{\nu_{1}/2}\nu_{2}^{\nu_{2}/2}x^{\nu_{1}/2-1} \\
& &
\times\left(\nu_{2}+\nu_{1}x\right)^{-\left(\nu_{1}+\nu_{2}\right)/2}
\frac{\Gamma\left(\frac{\nu_{1}}{2}\right)
\Gamma\left(1+\frac{\nu_{2}}{2}\right)
L_{\nu_{2}/2}^{\nu_{1}/2-1}
\left(-\frac{\lambda\nu_{1}x}
{2\left(\nu_{1}x+\nu_{2}\right)}\right)}
{B\left(\frac{\nu_{1}}{2},\frac{\nu_{2}}{2}\right)
\Gamma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)}
\end{eqnarray*}
where :math:`L_{\nu_{2}/2}^{\nu_{1}/2-1}(x)` is an associated Laguerre
polynomial.
If :math:`\lambda=0`, the distribution becomes equivalent to the Fisher
distribution with :math:`\nu_{1}` and :math:`\nu_{2}` degrees of freedom.
Implementation: `scipy.stats.ncf`
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