F distribution

Fisher-Snedecor

Probability density function
Cumulative distribution function
Parameters	d1, d2 > 0 deg. of freedom
Support	x ∈ [0, +∞)
PDF	${\displaystyle {\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!}$
CDF	${\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)}$
Mean	${\displaystyle {\frac {d_{2}}{d_{2}-2}}\!}$ for d2 > 2
Mode	${\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}}$ for d1 > 2
Variance	${\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!}$ for d2 > 4
Skewness	${\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!}$ for d2 > 6
Ex. kurtosis	see text
MGF	does not exist, raw moments defined in text and in
CF	see text

In probability theory and statistics, the F-distribution, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance, e.g., F-test.

If a random variable X has an F-distribution with parameters d₁ and d₂, we write X ~ F(d₁, d₂). Then the probability density function (pdf) for X is given by

for real x ≥ 0. Here ${\displaystyle \mathrm {B} }$ is the beta function. In many applications, the parameters d₁ and d₂ are positive integers, but the distribution is well-defined for positive real values of these parameters.

The cumulative distribution function is

where I is the regularized incomplete beta function.