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:

<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.

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

<math> 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} <math>

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

The cumulative distribution function is

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

where I is the regularized incomplete beta function.

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

mode 

<math> \frac{d_1-2}{d_1}\cdot\frac{d_2}{d_2+2} <math> provided d₁ > 2

mean 

<math>\mu = \frac{d_2}{d_2-2}<math> provided d₂ > 2

variance 

<math>\sigma^2 = \frac{2 d_2^2 (d_1 + d_2 - 2)}{d_1 (d_2-2)^2 (d_2-4)} <math> provided d₂ > 4

skewness 

<math>\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)}}<math> provided d₂ > 6

Generalization

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

External links

• Table of critical values of the F-distribution (http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm)
• Online significance testing with the F-distribution (http://home.clara.net/sisa/signhlp.htm)
• Distribution Calculator (http://www.vias.org/simulations/simusoft_distcalc.html) Calculates probabilities and critical values for normal, t-, chi2- and F-distribution

de:F-Verteilung it:Variabile casuale F di Snedecor nl:F-verdeling