F-distribution

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 R.A. 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 expectation, variance, and skewness are given in the sidebox; for $$d_2>8$$, the kurtosis is
 * $$\frac{12(20d_2-8d_2^2+d_2^3+44d_1-32d_1d_2+5d_2^2d_1-22d_1^2+5d_2d_1^2-16)}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}$$

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 &ge; 0, where d1 and d2 are positive integers, and B is the beta function.

The cumulative distribution function is


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

where I is the regularized incomplete beta function.

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

Related distributions and properties

 * $$Y \sim \chi^2$$ has a chi-square distribution if $$Y = \lim_{\nu_2 \to \infty} \nu_1 X$$ for $$X \sim \mathrm{F}(\nu_1, \nu_2)$$.
 * $$F(\nu_1,\nu_2)$$ is equivalent to the scaled Hotelling's T-square distribution $$(\nu_1(\nu_1+\nu_2-1)/\nu_2) T^2(\nu_1,\nu_1+\nu_2-1)$$.
 * One interesting property is that if $$X \sim F(\nu_1,\nu_2),\ \frac{1}{X} \sim F(\nu_2,\nu_1)$$.