F-distribution

Encyclopedia : F : FD : FDI : F-distribution

}}}\!\left(\frac,\frac\right)}\!]| cdf =[I_}(d_1/2, d_2/2)\!]| mean =[\frac\!] for [d_2 > 2]| median =| mode =[\frac\;\frac\!] for [d_1 > 2]| variance =[\frac\!] for [d_2 > 4]| skewness =[\frac}}\!]for [d_2 > 6]| kurtosis =see text| entropy =| mgf =see text for raw moments| char =| }} 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]

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]

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

[ g(x) = \frac(d_1/2, d_2/2)} \; \left(\frac\right)^ \; \left(1-\frac\right)^ \; x^ ]

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_}(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

[Y \sim \chi^2] is a chi-square distribution as [Y = \lim_ \nu_1 X] for [X \sim \mathrm(\nu_1, \nu_2)].

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

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