एफ वितरण

Fisher–Snedecor

Probability density function File:F-distribution pdf.svg
Cumulative distribution function File:F dist cdf.svg
Parameters	d1, d2 > 0 deg. of freedom
Support	$x\in (0,+\infty )\;$ if $d_{1}=1$, otherwise $x\in [0,+\infty )\;$
PDF	${\frac {\sqrt {\frac {(d_{1}x)^{d_{1}}d_{2}^{d_{2}}}{(d_{1}x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!$
CDF	$I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)$
Mean	${\frac {d_{2}}{d_{2}-2}}\!$ for d2 > 2
Mode	${\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}$ for d1 > 2
Variance	${\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!$ for d2 > 4
Skewness	${\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!$ for d2 > 6
Ex. kurtosis	see text
Entropy	$\ln \Gamma \left({\tfrac {d_{1}}{2}}\right)+\ln \Gamma \left({\tfrac {d_{2}}{2}}\right)-\ln \Gamma \left({\tfrac {d_{1}+d_{2}}{2}}\right)+\!$ $\left(1-{\tfrac {d_{1}}{2}}\right)\psi \left(1+{\tfrac {d_{1}}{2}}\right)-\left(1+{\tfrac {d_{2}}{2}}\right)\psi \left(1+{\tfrac {d_{2}}{2}}\right)\!$ $+\left(\frac{d_1+d_2}{2}\right)\mathrm{\psi <!--\; \psi \; -->}\left(\frac{d_1+d_2}{2}\right)+\ln <!--\; \; -->\frac{d_1}{{}_{}}$