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 Ronald Fisher and George W. Snedecor).

A random variate of the F-distribution arises as the ratio of two chi-squared variates:

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

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

The cumulative distribution function is

where I is the regularized incomplete beta function.

Generalization

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

External links

• Table of critical values of the F-distribution
• Online significance testing with the F-distribution
• Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution