F-distribution

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Fisher-Snedecor
Probability density function
Cumulative distribution function
Parameters	$d_1>0,\ d_2>0$ deg. of freedom
Support	$x \in [0; +\infty)\!$
Probability density function (pdf)	$\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!$
Cumulative distribution function (cdf)	$I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)\!$
Mean	$\frac{d_2}{d_2-2}\!$ for $d 2 > 2$
Median
Mode	$\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!$ for $d 1 > 2$
Variance	$\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!$ for $d 2 > 4$
Skewness	$\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\!$ for $d 2 > 6$
Excess kurtosis	see text
Entropy
Moment-generating function (mgf)	see text for raw moments
Characteristic function

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:

$\frac{U_1/d_1}{U_2/d_2}$

where

U₁ and U₂ have chi-square distributions with d₁ and d₂ degrees of freedom respectively, and

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

$\frac{12(20d_2-8d_2^2+d_2^3+44d_1-32d_1d_2+5d_2^2d_1-22d_1^2+5d_2d_1^2-16)}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}$

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.

[edit] Generalization

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

[edit] Related distributions and properties

$Y \sim \chi^2$ has a chi-square distribution if $Y = \lim_{\nu_2 \to \infty} \nu_1 X$ for $X \sim \mathrm{F}(\nu_1, \nu_2)$ .
$F (ν 1,ν 2)$ is equivalent to the scaled Hotelling's T-square distribution $(ν 1 (ν 1 + ν 2 - 1) / ν 2) T 2 (ν 1,ν 1 + ν 2 - 1)$ .
One interesting property is that if $X \sim F(\nu_1,\nu_2),\ \frac{1}{X} \sim F(\nu_2,\nu_1)$ .

[edit] 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
Cumulative density function (CDF) calculator for the Fisher F-distribution
Probability density function (PDF) calculator for the Fisher F-distribution

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