Editing F-distribution (section)

==Properties==
The expectation, variance, and other details about the F(''d''<sub>1</sub>, ''d''<sub>2</sub>) are given in the sidebox; for ''d''<sub>2</sub>&nbsp;>&nbsp;8, the [[excess kurtosis]] is

<math display="block">\gamma_2 = 12\frac{d_1(5d_2-22)(d_1+d_2-2)+(d_2-4)(d_2-2)^2}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}.</math>

The ''k''-th moment of an F(''d''<sub>1</sub>, ''d''<sub>2</sub>) distribution exists and is finite only when 2''k'' < ''d''<sub>2</sub> and it is equal to<ref name=taboga>{{cite web | last1 = Taboga | first1 = Marco | url = http://www.statlect.com/F_distribution.htm | title = The F distribution}}</ref>

<math display="block">\mu _X(k) =\left( \frac{d_2}{d_1}\right)^k \frac{\Gamma \left(\tfrac{d_1}{2}+k\right) }{\Gamma \left(\tfrac{d_1}{2}\right)} \frac{\Gamma \left(\tfrac{d_2}{2}-k\right) }{\Gamma \left( \tfrac{d_2}{2}\right) }.</math>

The ''F''-distribution is a particular [[Parametrization (geometry)|parametrization]] of the [[beta prime distribution]], which is also called the beta distribution of the second kind.

The [[Characteristic function (probability theory)|characteristic function]] is listed incorrectly in many standard references (e.g.,<ref name=abramowitz />). The correct expression <ref>Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," ''[[Biometrika]]'', 69: 261–264 {{JSTOR|2335882}}</ref> is

<math display="block">\varphi^F_{d_1, d_2}(s) = \frac{\Gamma{\left(\frac{d_1+d_2}{2}\right)}}{\Gamma{\left(\tfrac{d_2}{2}\right)}} U \! \left(\frac{d_1}{2},1-\frac{d_2}{2},-\frac{d_2}{d_1} \imath s \right)</math>

where ''U''(''a'', ''b'', ''z'') is the [[confluent hypergeometric function]] of the second kind.