F-distribution

Fisher-Snedecor

Probability density function
Cumulative distribution function
Parameters	<math>d_1>0,\ d_2>0</math> deg. of freedom
Support	<math>x \in [0; +\infty)\!</math>
Probability density function (pdf)	<math>\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2
Cumulative distribution function (cdf)	{{{cdf}}}
Mean	{{{mean}}}
Median	{{{median}}}
Mode	{{{mode}}}
Variance	{{{variance}}}
Skewness	{{{skewness}}}
Excess kurtosis	{{{kurtosis}}}
Entropy	{{{entropy}}}
Moment-generating function (mgf)	{{{mgf}}}
Characteristic function	{{{char}}}

{(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!</math>|

 cdf        =<math>I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)\!</math>|
 mean       =<math>\frac{d_2}{d_2-2}\!</math> for <math>d_2 > 2</math>|
 median     =|
 mode       =<math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!</math> for <math>d_1 > 2</math>|
 variance   =<math>\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!</math> for <math>d_2 > 4</math>|
 skewness   =<math>\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\!</math>
for <math>d_2 > 6</math>|
 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:

<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 expectation, variance, and skewness are given in the sidebox; for <math>d_2>8</math>, the kurtosis is

<math>\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)}.</math>

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.

Generalization

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

Related distributions and properties

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

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