F-distribution

Probability distribution
name =Fisher-Snedecor
type =density
pdf_

cdf_

parameters =$d\_1>0,\; d\_2>0$ deg. of freedom
support =$x\; in\; [0,\; +infty)!$
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} ight)}!
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 $d\_2\; >\; 6$
kurtosis ="see text"
entropy =
mgf ="does not exist, raw moments defined elsewhere"
char ="defined elsewhere"
In probability theory and statistics, the "F"-distribution is a continuous probability distribution.Abramowitz_Stegun_ref|26|946] [NIST (2006). [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3665.htm Engineering Statistics Handbook - F Distribution] ] [cite book
last = Mood
first = Alexander
coauthors = Franklin A. Graybill, Duane C. Boes
title = Introduction to the Theory of Statistics (Third Edition, p. 246-249)
publisher = McGraw-Hill
date = 1974
isbn = 0-07-042864-6] 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

*"U"_"1" and "U"₂ have chi-square distributions with "d"_"1" and "d"₂ degrees of freedom respectively, and

*"U"₁ and "U"₂ are independent (see Cochran's theorem for an application).

The probability density function of an "F"("d"₁, "d"₂) distributed random variable is given by

:$f(x)\; =\; frac\{left(frac\{d\_1,x\}\{d\_1,x\; +\; d\_2\}\; ight)^\{d\_1/2\}\; ;\; left(1-frac\{d\_1,x\}\{d\_1,x\; +\; d\_2\}\; ight)^\{d\_2/2\{x;\; mathrm\{B\}(d\_1/2,\; d\_2/2)\}$

for real "x" ≥ 0, where "d"₁ and "d"₂ are positive integers, and B is the beta function.

The cumulative distribution function is $F(x)=I\_\{frac\{d\_1\; x\}\{d\_1\; x\; +\; d\_2(d\_1/2,\; d\_2/2)$

where "I" is the regularized incomplete beta function.

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 $A=5d\_2^2d\_1-22d\_1^2+5d\_2d\_1^2-16.$

Generalization

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

Related distributions and properties

*If $X\; sim\; mathrm\{F\}(\; u\_1,\; u\_2)$ then $Y\; =\; lim\_\{\; u\_2\; o\; infty\}\; u\_1\; X$ has the chi-square distribution $chi^2\_\{\; u\_\{1$
*$operatorname\{F\}(\; u\_1,\; u\_2)$ is equivalent to the scaled Hotelling's T-square distribution $(\; u\_1(\; u\_1+\; u\_2-1)/\; u\_2)operatorname\{T\}^2(\; u\_1,\; u\_1+\; u\_2-1)$.
*If $X\; sim\; operatorname\{F\}(\; u\_1,\; u\_2),$ then $frac\{1\}\{X\}\; sim\; F(\; u\_2,\; u\_1)$.
*if $X\; sim\; mathrm\{t\}(\; u)!$ has a Student's t-distribution then $X^2\; sim\; operatorname\{F\}(\; u\_1\; =\; 1,\; u\_2\; =\; u)$.
*if $X\; sim\; operatorname\{F\}(\; u\_1,\; u\_2)$ and $Y=frac\{\; u\_1\; X/\; u\_2\}\{1+\; u\_1\; X/\; u\_2\}$ then $Y\; sim\; operatorname\{Beta\}(\; u\_1/2,\; u\_2/2)$ has a Beta-distribution.
*if $operatorname\{Q\}\_X(p)$ is the quantile $p$ for $Xsim\; operatorname\{F\}(\; u\_1,\; u\_2)$ and $operatorname\{Q\}\_Y(p)$ is the quantile $p$ for $Ysim\; operatorname\{F\}(\; u\_2,\; u\_1)$ then $operatorname\{Q\}\_X(p)=1/operatorname\{Q\}\_Y(p)$.

References

External links

* [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm Table of critical values of the "F"-distribution]
* [http://home.clara.net/sisa/signhlp.htm Online significance testing with the F-distribution]
* [http://www.vias.org/simulations/simusoft_distcalc.html Distribution Calculator] Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
* [http://www.danielsoper.com/statcalc/calc39.aspx Cumulative distribution function (CDF) calculator for the Fisher F-distribution]
* [http://www.danielsoper.com/statcalc/calc38.aspx Probability density function (PDF) calculator for the Fisher F-distribution]