F-distribution

F-distribution

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In probability theory and statistics, the F-distribution is a continuous probability distribution.^[1]^[2]^[3] It is also known as Snedecor's F distribution or the Fisher-Snedecor distribution (after R.A. Fisher and George W. Snedecor). 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.

Characterization

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

f(x) = \frac{\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\; \mathrm{B}(d_1/2, d_2/2)}

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

The cumulative distribution function is F(x)=I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)

The expectation, variance, and other details about the F(d_1,d_2) are given in the sidebox; for d_2>8, the kurtosis is

\frac{20d_2-8d_2^2+d_2^3+44d_1-32d_1d_2+A}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)/12}

where A=5d_2^2d_1-22d_1^2+5d_2d_1^2-16.

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

If X \sim \mathrm{F}(\nu_1, \nu_2) then Y = \lim_{\nu_2 \to \infty} \nu_1 X has the chi-square distribution \chi^2_{\nu_{1}}
\operatorname{F}(\nu_1,\nu_2) is equivalent to the scaled Hotelling's T-square distribution (\nu_1(\nu_1+\nu_2-1)/\nu_2)\operatorname{T}^2(\nu_1,\nu_1+\nu_2-1).
If X \sim \operatorname{F}(\nu_1,\nu_2), then \frac{1}{X} \sim F(\nu_2,\nu_1).
if X \sim \mathrm{t}(\nu)\! has a Student's t-distribution then X^2 \sim \operatorname{F}(\nu_1 = 1, \nu_2 = \nu).
if X \sim \operatorname{F}(\nu_1,\nu_2) and Y=\frac{\nu_1 X/\nu_2}{1+\nu_1 X/\nu_2} then Y \sim \operatorname{Beta}(\nu_1/2,\nu_2/2) has a Beta-distribution.
if \operatorname{Q}_X(p) is the quantile p for X\sim \operatorname{F}(\nu_1,\nu_2) and \operatorname{Q}_Y(p) is the quantile p for Y\sim \operatorname{F}(\nu_2,\nu_1) then \operatorname{Q}_X(p)=1/\operatorname{Q}_Y(p).