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F-distribution

In statistics, the F-distribution is the probability distribution of a random variable of the form

<math>\frac{U_1/d_1}{U_2/d_2}<math>

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.

If d₂ > 2, then the mean is d₂/(d₂ − 2) and if d₂ > 4 the variance is given by

<math>

\frac{2d_2^2(d_1+d_2-2)}{d_1(d_2-2)^2(d_2-4)}. <math>

The F-distribution always ranges from zero to infinity; if d₁ > 2, it is a unimodal with mode

<math>

\frac{d_1-2}{d_1}\cdot\frac{d_2}{d_2+1} <math>