Editing F-distribution (section)

==Definitions==
The ''F''-distribution with ''d''<sub>1</sub> and ''d''<sub>2</sub> degrees of freedom is the distribution of

<math display="block"> X = \frac{U_1/d_1}{U_2/d_2} </math>

where <math display=inline>U_1</math> and <math display=inline>U_2</math> are [[Independence (probability theory)|independent]] [[random variable]]s with [[chi-square distribution]]s with respective degrees of freedom <math display=inline>d_1</math> and <math display=inline>d_2</math>.

It can be shown to follow that the [[probability density function]] (pdf) for ''X'' is given by

<math display="block">\begin{align}
f(x; d_1,d_2) &= \frac{\sqrt{\frac{(d_1x)^{d_1}\,\,d_2^{d_2}} {(d_1x+d_2)^{d_1+d_2}}}} {x\operatorname{B}\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \\[5pt]
&=\frac{1}{\operatorname{B}\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{\frac{d_1}{2}} x^{\frac{d_1}{2} - 1} \left(1+\frac{d_1}{d_2} \, x \right)^{-\frac{d_1+d_2}{2}}
\end{align}</math>

for [[real number|real]] ''x'' > 0. Here <math>\mathrm{B}</math> is the [[beta function]]. In many applications, the parameters ''d''<sub>1</sub> and ''d''<sub>2</sub> are [[positive integer]]s, but the distribution is well-defined for positive real values of these parameters.

The [[cumulative distribution function]] is

<math display="block">F(x; d_1,d_2)=I_{d_1 x/(d_1 x + d_2)}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) ,</math>

where ''I'' is the [[regularized incomplete beta function]].