f-distribution

In statistics and probability, 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

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. 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. An F(d₁, d₂) random variable has the following properties:

mode: $\frac{d_1-2}{d_1}\cdot\frac{d_2}{d_2+2}$ provided d₁ > 2
mean: $\mu = \frac{d_2}{d_2-2}$ provided d₂ > 2
variance: $\sigma^2 = \frac{2 d_2^2 (d_1 + d_2 - 2)}{d_1 (d_2-2)^2 (d_2-4)}$ provided d₂ > 4
skewness: $\gamma_1 = \frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}$ provided d₂ > 6

Generalization

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

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