Wikia

**Fisher-Snedecor**
Probability density function None uploaded yet.
Cumulative distribution function None uploaded yet.
Parameters	$d_1>0,\ d_2>0$ deg. of freedom
Support	$x \in [0; +\infty)\!$
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	$I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)\!$
Mean	$\frac{d_2}{d_2-2}\!$ for $d_2 > 2$
Median
Mode	$\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!$ for $d_1 > 2$
Variance	$\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!$ for $d_2 > 4$
Skewness	$\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\!$ for $d_2 > 6$
Kurtosis
Entropy
mgf	see text for raw moments
Char. func.

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 Ronald Fisher and George W. Snedecor).

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

$\frac{U_1/d_1}{U_2/d_2}$

where

U₁ and U₂ have chi-square distributions with d₁ and d₂ degrees of freedom respectively, and

U₁ and U₂ are independent (see Cochran's theorem for an application).

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

$g(x) = \frac{1}{\mathrm{B}(d_1/2, d_2/2)} \; \left(\frac{d_1\,x}{d_1\,x + d_2}\right)^{d_1/2} \; \left(1-\frac{d_1\,x}{d_1\,x + d_2}\right)^{d_2/2} \; x^{-1}$