# F-distribution

Parameters Probability density function325px Cumulative distribution function325px ${\displaystyle d_{1}>0,\ d_{2}>0}$ deg. of freedom ${\displaystyle x\in [0;+\infty )\!}$ ${\displaystyle {\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!}$ ${\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}(d_{1}/2,d_{2}/2)\!}$ ${\displaystyle {\frac {d_{2}}{d_{2}-2}}\!}$ for ${\displaystyle d_{2}>2}$ ${\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}\!}$ for ${\displaystyle d_{1}>2}$ ${\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!}$ for ${\displaystyle d_{2}>4}$ ${\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!}$for ${\displaystyle d_{2}>6}$ see text see text for raw moments

In probability theory and statistics, the F-distribution is a continuous probability distribution. It is also known as Snedecor's F distribution or the Fisher-Snedecor distribution (after R.A. Fisher and George W. Snedecor).

A random variate of the F-distribution arises as the ratio of two chi-squared variates:

${\displaystyle {\frac {U_{1}/d_{1}}{U_{2}/d_{2}}}}$

where

The F-distribution arises frequently as the null distribution of a test statistic, especially in likelihood-ratio tests, perhaps most notably in the analysis of variance; see F-test.

The expectation, variance, and skewness are given in the sidebox; for ${\displaystyle d_{2}>8}$, the kurtosis is

${\displaystyle {\frac {12(20d_{2}-8d_{2}^{2}+d_{2}^{3}+44d_{1}-32d_{1}d_{2}+5d_{2}^{2}d_{1}-22d_{1}^{2}+5d_{2}d_{1}^{2}-16)}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}}$

The probability density function of an F(d1, d2) distributed random variable is given by

${\displaystyle g(x)={\frac {1}{\mathrm {B} (d_{1}/2,d_{2}/2)}}\;\left({\frac {d_{1}\,x}{d_{1}\,x+d_{2}}}\right)^{d_{1}/2}\;\left(1-{\frac {d_{1}\,x}{d_{1}\,x+d_{2}}}\right)^{d_{2}/2}\;x^{-1}}$

for real x ≥ 0, where d1 and d2 are positive integers, and B is the beta function.

${\displaystyle G(x)=I_{\frac {d_{1}x}{d_{1}x+d_{2}}}(d_{1}/2,d_{2}/2)}$

where I is the regularized incomplete beta function.

## Generalization

A generalization of the (central) F-distribution is the noncentral F-distribution.

## Related distributions and properties

• ${\displaystyle Y\sim \chi ^{2}}$ has a chi-square distribution if ${\displaystyle Y=\lim _{\nu _{2}\to \infty }\nu _{1}X}$ for ${\displaystyle X\sim \mathrm {F} (\nu _{1},\nu _{2})}$.
• ${\displaystyle F(\nu _{1},\nu _{2})}$ is equivalent to the scaled Hotelling's T-square distribution ${\displaystyle (\nu _{1}(\nu _{1}+\nu _{2}-1)/\nu _{2})T^{2}(\nu _{1},\nu _{1}+\nu _{2}-1)}$.
• One interesting property is that if ${\displaystyle X\sim F(\nu _{1},\nu _{2}),\ {\frac {1}{X}}\sim F(\nu _{2},\nu _{1})}$.