# F-distribution

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 Probability density functionNone uploaded yet. Cumulative distribution functionNone 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

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

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

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

$G(x) = I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)$

where I is the regularized incomplete beta function.

## Generalization

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

## Related distributions

• $Y \sim \chi^2$ is a chi-square distribution as $Y = \lim_{\nu_2 \to \infty} \nu_1 X$ for $X \sim \mathrm{F}(\nu_1, \nu_2)$.