F-distribution

Fisher–Snedecor
Probability density function
Cumulative distribution function
Parameters	d1, d2 > 0 deg. of freedom
Support	${\displaystyle x\in (0,+\infty )\;}$ if ${\displaystyle d_{1}=1}$, otherwise ${\displaystyle x\in [0,+\infty )\;}$
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
Excess kurtosis	see text
Entropy	${\displaystyle {\begin{aligned}&\ln \Gamma {\left({\tfrac {d_{1}}{2}}\right)}+\ln \Gamma {\left({\tfrac {d_{2}}{2}}\right)}-\ln \Gamma {\left({\tfrac {d_{1}+d_{2}}{2}}\right)}\\&+\left(1-{\tfrac {d_{1}}{2}}\right)\psi {\left(1+{\tfrac {d_{1}}{2}}\right)}-\left(1+{\tfrac {d_{2}}{2}}\right)\psi {\left(1+{\tfrac {d_{2}}{2}}\right)}\\&+\left({\tfrac {d_{1}+d_{2}}{2}}\right)\psi {\left({\tfrac {d_{1}+d_{2}}{2}}\right)}+\ln {\frac {d_{2}}{d_{1}}}\end{aligned}}}$
MGF	does not exist, raw moments defined in text and in
CF	see text

In probability theory and statistics, the F-distribution or F-ratio, 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 (ANOVA) and other F-tests.