Editing Noncentral F-distribution (section)

== Occurrence and specification ==
If <math>X</math> is a [[Noncentral chi-squared distribution|noncentral chi-squared]] random variable with noncentrality parameter <math>\lambda</math> and <math>\nu_1</math> degrees of freedom, and <math>Y</math> is a [[Chi-squared distribution|chi-squared]] random variable with <math>\nu_2</math> degrees of freedom that is [[statistical independence|statistically independent]] of <math>X</math>, then
:<math>
F=\frac{X/\nu_1}{Y/\nu_2}
</math>
is a noncentral ''F''-distributed random variable.
The [[probability density function]] (pdf) for the noncentral ''F''-distribution is<ref>{{cite book |first=S. |last=Kay |title=Fundamentals of Statistical Signal Processing: Detection Theory |location=New Jersey |publisher=Prentice Hall |year=1998 |page=29 |isbn=0-13-504135-X }}</ref>
:<math>
p(f)
=\sum\limits_{k=0}^\infty\frac{e^{-\lambda/2}(\lambda/2)^k}{ B\left(\frac{\nu_2}{2},\frac{\nu_1}{2}+k\right) k!}
\left(\frac{\nu_1}{\nu_2}\right)^{\frac{\nu_1}{2}+k}
\left(\frac{\nu_2}{\nu_2+\nu_1f}\right)^{\frac{\nu_1+\nu_2}{2}+k}f^{\nu_1/2-1+k}
 </math>
when <math>f\ge0</math> and zero otherwise.
The degrees of freedom <math>\nu_1</math> and <math>\nu_2</math> are positive.
The term <math>B(x,y)</math> is the [[beta function]], where
:<math>
B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.
</math>

The [[cumulative distribution function]] for the noncentral ''F''-distribution is
:<math>
F(x\mid d_1,d_2,\lambda)=\sum\limits_{j=0}^\infty\left(\frac{\left(\frac{1}{2}\lambda\right)^j}{j!}e^{-\lambda/2} \right)I\left(\frac{d_1x}{d_2 + d_1x}\bigg|\frac{d_1}{2}+j,\frac{d_2}{2}\right)
</math>
where <math>I</math> is the [[regularized incomplete beta function]].

The mean and variance of the noncentral ''F''-distribution are
:<math>
\operatorname{E}[F] \quad
\begin{cases}
= \frac{\nu_2(\nu_1+\lambda)}{\nu_1(\nu_2-2)} & \text{if } \nu_2>2\\
\text{does not exist} & \text{if } \nu_2\le2\\
\end{cases}
</math>
and
:<math>
\operatorname{Var}[F] \quad
\begin{cases}
= 2\frac{(\nu_1+\lambda)^2+(\nu_1+2\lambda)(\nu_2-2)}{(\nu_2-2)^2(\nu_2-4)}\left(\frac{\nu_2}{\nu_1}\right)^2
& \text{if } \nu_2>4\\
\text{does not exist}
& \text{if } \nu_2\le4.\\
\end{cases}
</math>