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]