View source for F-distribution

{{About|the central F-distribution|the generalized distribution|noncentral F-distribution|}}
{{distinguish|text=[[F-statistics|''F''-statistics]] as used in [[population genetics]]}}
{{DISPLAYTITLE:''F''-distribution}}
{{Probability distribution
| name       = Fisher–Snedecor
| type       = density
| pdf_image  = [[Image:F-distribution pdf.svg|325px]]|
| cdf_image  = [[Image:F_dist_cdf.svg|325px]]|
| parameters = ''d''<sub>1</sub>, ''d''<sub>2</sub> > 0 deg. of freedom|
| support    = <math>x \in (0, +\infty)\;</math> if <math>d_1 = 1</math>, otherwise <math>x \in [0, +\infty)\;</math> | 
| pdf        = <math>\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)}\!</math>
| cdf        = <math>I_{\frac{d_1 x}{d_1 x + d_2}} \left(\tfrac{d_1}{2}, \tfrac{d_2}{2} \right)</math>
| mean       = <math>\frac{d_2}{d_2-2}\!</math><br /> for ''d''<sub>2</sub> > 2
| median     =
| mode       = <math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}</math><br /> for ''d''<sub>1</sub> > 2
| variance   = <math>\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!</math><br /> for ''d''<sub>2</sub> > 4
| 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><br />for ''d''<sub>2</sub> > 6
| kurtosis   = ''see text''
| entropy    = <math>\ln \Gamma \left(\tfrac{d_1}{2} \right) + \ln \Gamma \left(\tfrac{d_2}{2} \right) - \ln \Gamma \left(\tfrac{d_1+d_2}{2} \right) + \!</math><br /><math>\left(1-\tfrac{d_1}{2} \right) \psi \left(1+\tfrac{d_1}{2} \right) - \left(1+\tfrac{d_2}{2} \right) \psi \left(1+\tfrac{d_2}{2} \right) \!</math><br /><math> + \left(\tfrac{d_1 + d_2}{2} \right) \psi \left(\tfrac{d_1 + d_2}{2} \right) + \ln \frac{d_1}{d_2} \!</math><ref name=lazo1978entropy>{{Cite journal|last=Lazo |first=A.V. |last2=Rathie |first2=P. |title=On the entropy of continuous probability distributions |journal=IEEE Transactions on Information Theory |volume=24 |number=1 |pages=120–122 |year=1978 |publisher=IEEE |doi=10.1109/tit.1978.1055832}}</ref>
| mgf        = ''does not exist, raw moments defined in text and in <ref name=johnson /><ref name=abramowitz />''
| char       = ''see text''
}}

In [[probability theory]] and [[statistics]], the '''''F''-distribution''', also known as '''Snedecor's ''F'' distribution''' or the '''Fisher–Snedecor distribution''' (after [[Ronald Fisher]] and [[George W. Snedecor]]) is a [[continuous probability distribution]] that arises frequently as the [[null distribution]] of a [[test statistic]], most notably in the [[analysis of variance]] (ANOVA), e.g., [[F-test|''F''-test]].{{clarify |date=June 2019 |reason=ANOVA uses the F-test, but the two are not synonymous, right? }}<ref name=johnson>{{cite book  | last = Johnson
  | first = Norman Lloyd
  |author2=Samuel Kotz |author3=N. Balakrishnan
   | title = Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27)
  | publisher = Wiley
  | year = 1995
  | isbn = 0-471-58494-0}}</ref><ref name=abramowitz>{{Abramowitz_Stegun_ref|26|946}}</ref><ref>NIST (2006).  [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3665.htm Engineering Statistics Handbook – F Distribution]</ref><ref>{{cite book  | last = Mood
  | first = Alexander
  |author2=Franklin A. Graybill |author3=Duane C. Boes
   | title = Introduction to the Theory of Statistics (Third Edition, pp. 246–249)
  | publisher = McGraw-Hill
  | year = 1974
  | isbn = 0-07-042864-6}}</ref>

==Definition==
If a [[random variable]] ''X'' has an ''F''-distribution with parameters ''d''<sub>1</sub> and ''d''<sub>2</sub>, we write ''X'' ~ F(''d''<sub>1</sub>, ''d''<sub>2</sub>). Then the [[probability density function]] (pdf) for ''X'' is given by

<math> 
\begin{align}
f(x; d_1,d_2) &= \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)} \\
&=\frac{1}{\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{\frac{d_1}{2}} x^{\frac{d_1}{2} - 1} \left(1+\frac{d_1}{d_2}\,x\right)^{-\frac{d_1+d_2}{2}}
\end{align}
</math>

for [[real number|real]] ''x'' > 0. Here <math>\mathrm{B}</math> is the [[beta function]]. In many applications, the parameters ''d''<sub>1</sub> and ''d''<sub>2</sub> are [[positive integer]]s, but the distribution is well-defined for positive real values of these parameters.

The [[cumulative distribution function]] is

:<math>F(x; d_1,d_2)=I_{\frac{d_1 x}{d_1 x + d_2}}\left (\tfrac{d_1}{2}, \tfrac{d_2}{2} \right) ,</math>

where ''I'' is the [[regularized incomplete beta function]].

The expectation, variance, and other details about the F(''d''<sub>1</sub>, ''d''<sub>2</sub>) are given in the sidebox; for ''d''<sub>2</sub> > 8, the [[excess kurtosis]] is

:<math>\gamma_2 = 12\frac{d_1(5d_2-22)(d_1+d_2-2)+(d_2-4)(d_2-2)^2}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}.</math>

The ''k''-th moment of an F(''d''<sub>1</sub>, ''d''<sub>2</sub>) distribution exists and is finite only when 2''k'' < ''d''<sub>2</sub> and it is equal to <ref name=taboga>{{cite web | last1 = Taboga | first1 = Marco | url = http://www.statlect.com/F_distribution.htm | title = The F distribution}}</ref>

:<math>\mu _{X}(k) =\left( \frac{d_{2}}{d_{1}}\right)^{k}\frac{\Gamma \left(\tfrac{d_1}{2}+k\right) }{\Gamma \left(\tfrac{d_1}{2}\right) }\frac{\Gamma \left(\tfrac{d_2}{2}-k\right) }{\Gamma \left( \tfrac{d_2}{2}\right) }</math>

The ''F''-distribution is a particular parametrization of the [[beta prime distribution]], which is also called the beta distribution of the second kind.

The [[Characteristic function (probability theory)|characteristic function]] is listed incorrectly in many standard references (e.g.,<ref name=abramowitz />). The correct expression <ref>Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," ''[[Biometrika]]'', 69: 261–264  {{jstor|2335882}}</ref> is

:<math>\varphi^F_{d_1, d_2}(s) = \frac{\Gamma(\frac{d_1+d_2}{2})}{\Gamma(\tfrac{d_2}{2})} U \! \left(\frac{d_1}{2},1-\frac{d_2}{2},-\frac{d_2}{d_1} \imath s \right)</math>

where ''U''(''a'', ''b'', ''z'') is the [[confluent hypergeometric function]] of the second kind.

==Characterization==
A [[random variate]] of the ''F''-distribution with parameters ''d''<sub>1</sub> and ''d''<sub>2</sub> arises as the ratio of two appropriately scaled [[chi-squared distribution|chi-squared]] variates:<ref>M.H. DeGroot (1986), ''Probability and Statistics'' (2nd Ed), Addison-Wesley. {{ISBN|0-201-11366-X}}, p. 500</ref>

:<math>X = \frac{U_1/d_1}{U_2/d_2}</math>

where

*''U''<sub>1</sub> and ''U''<sub>2</sub> have [[chi-squared distribution]]s with ''d''<sub>1</sub> and ''d''<sub>2</sub> [[Degrees of freedom (statistics)|degrees of freedom]] respectively, and
*''U''<sub>1</sub> and ''U''<sub>2</sub> are [[statistical independence|independent]].

In instances where the ''F''-distribution is used, for example in the [[analysis of variance]], independence of ''U''<sub>1</sub> and ''U''<sub>2</sub> might be demonstrated by applying [[Cochran's theorem]].

Equivalently, the random variable of the ''F''-distribution may also be written

:<math> \left. X = \frac{s_1^2}{\sigma_1^2} \right/ \frac{s_2^2}{\sigma_2^2}</math>

where ''s''<sub>1</sub><sup>2</sup> and ''s''<sub>2</sub><sup>2</sup> are the sums of squares ''S''<sub>1</sub><sup>2</sup> and ''S''<sub>2</sub><sup>2</sup> from two normal processes with variances σ<sub>1</sub><sup>2</sup> and σ<sub>2</sub><sup>2</sup> divided by the corresponding number of χ<sup>2</sup> degrees of freedom, ''d''<sub>1</sub> and ''d''<sub>2</sub> respectively : <math>s_1^2 = \frac{S_1^2}{d_1}</math> and <math>s_2^2 = \frac{S_2^2}{d_2}</math>.{{discuss|Inconsistent.2C_or_at_least_confusing.2C_representation_in_terms_of_normal_variables}}{{citation needed|date=March 2014}}

In a [[frequentist]] context, a scaled ''F''-distribution therefore gives the probability ''p''(''s''<sub>1</sub><sup>2</sup>/''s''<sub>2</sub><sup>2</sup> | σ<sub>1</sub><sup>2</sup>, σ<sub>2</sub><sup>2</sup>), with the ''F''-distribution itself, without any scaling, applying where σ<sub>1</sub><sup>2</sup> is being taken equal to σ<sub>2</sub><sup>2</sup>.  This is the context in which the ''F''-distribution most generally appears in [[F-test|''F''-tests]]: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis.

The quantity ''X'' has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant [[Jeffreys prior]] is taken for the [[prior probability|prior probabilities]] of σ<sub>1</sub><sup>2</sup> and σ<sub>2</sub><sup>2</sup>.<ref>G. E. P. Box and G. C. Tiao (1973), ''Bayesian Inference in Statistical Analysis'', Addison-Wesley. p. 110</ref> In this context, a scaled ''F''-distribution thus gives the posterior probability ''p''(σ<sub>2</sub><sup>2</sup>/σ<sub>1</sub><sup>2</sup>|''s''<sub>1</sub><sup>2</sup>, ''s''<sub>2</sub><sup>2</sup>), where now the observed sums ''s''<sub>1</sub><sup>2</sup> and ''s''<sub>2</sub><sup>2</sup> are what are taken as known.

== {{anchor|Properties}} Properties and related distributions ==
*If <math>X \sim \chi^2_{d_1}</math> and <math>Y \sim \chi^2_{d_2}</math> are [[independence (probability theory)|independent]], then <math> \frac{X / d_1}{Y / d_2} \sim \mathrm{F}(d_1, d_2)</math>
*If <math>X_k \sim \Gamma(\alpha_k,\beta_k)\,</math> are independent, then <math> \frac{\alpha_2\beta_1 X_1}{\alpha_1\beta_2 X_2} \sim \mathrm{F}(2\alpha_1, 2\alpha_2)</math>
*If <math>X \sim \operatorname{Beta}(d_1/2,d_2/2)</math> ([[Beta distribution]]) then <math>\frac{d_2 X}{d_1(1-X)} \sim \operatorname{F}(d_1,d_2)</math>
*Equivalently, if ''X'' ~ F(''d''<sub>1</sub>, ''d''<sub>2</sub>), then <math>\frac{d_1 X/d_2}{1+d_1 X/d_2} \sim \operatorname{Beta}(d_1/2,d_2/2)</math>.
*If ''X'' ~ F(''d''<sub>1</sub>, ''d''<sub>2</sub>), then <math>\frac{d_1}{d_2}X</math> has a [[beta prime distribution]]: <math>\frac{d_1}{d_2}X \sim \operatorname{\beta^\prime}(\tfrac{d_1}{2},\tfrac{d_2}{2})</math>.
*If ''X'' ~ F(''d''<sub>1</sub>, ''d''<sub>2</sub>) then <math>Y = \lim_{d_2 \to \infty} d_1 X</math> has the [[chi-squared distribution]] <math>\chi^2_{d_1}</math>
*F(''d''<sub>1</sub>, ''d''<sub>2</sub>) is equivalent to the scaled [[Hotelling's T-squared distribution]] <math>\frac{d_2}{d_1(d_1+d_2-1)} \operatorname{T}^2 (d_1, d_1 +d_2-1) </math>.
*If ''X'' ~ F(''d''<sub>1</sub>, ''d''<sub>2</sub>) then ''X''<sup>−1</sup> ~ F(''d''<sub>2</sub>, ''d''<sub>1</sub>).
*If ''X'' ~ t(''n'') — [[Student's t-distribution]] — then:
::<math>X^{2} \sim \operatorname{F}(1, n) </math>
::<math>X^{-2} \sim \operatorname{F}(n, 1) </math>
*''F''-distribution is a special case of type 6 [[Pearson distribution]]
*If ''X'' and ''Y'' are independent, with ''X'', ''Y'' ~ [[Laplace distribution|Laplace(''μ'', ''b'')]] then
::<math> \frac{|X-\mu|}{|Y-\mu|} \sim \operatorname{F}(2,2) </math>
*If ''X'' ~ F(''n'', ''m'') then <math>\tfrac{\log{X}}{2} \sim \operatorname{FisherZ}(n,m)</math> ([[Fisher's z-distribution]])
*The [[noncentral F-distribution|noncentral ''F''-distribution]] simplifies to the ''F''-distribution if λ = 0.
*The doubly [[noncentral F-distribution|noncentral ''F''-distribution]] simplifies to the ''F''-distribution if <math> \lambda_1 = \lambda_2 = 0 </math>
*If <math>\operatorname{Q}_X(p)</math> is the quantile ''p'' for ''X'' ~ F(''d''<sub>1</sub>, ''d''<sub>2</sub>) and <math>\operatorname{Q}_Y(1-p)</math> is the quantile 1&nbsp;−&nbsp;''p'' for ''Y'' ~ F(''d''<sub>2</sub>, ''d''<sub>1</sub>), then
::<math>\operatorname{Q}_X(p)=\frac{1}{\operatorname{Q}_Y(1-p)}.</math>

== See also ==
{{Colbegin}}
* [[Beta prime distribution]]
* [[Chi-squared distribution]]
* [[Chow test]]
* [[Gamma distribution]]
* [[Hotelling's T-squared distribution]]
* [[Wilks' lambda distribution]]
* [[Wishart distribution]]
{{Colend}}

== References ==
{{reflist}}

== External links ==
*[http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm Table of critical values of the ''F''-distribution]
*[http://jeff560.tripod.com/f.html Earliest Uses of Some of the Words of Mathematics: entry on ''F''-distribution contains a brief history]
*[http://www.waterlog.info/f-test.htm Free calculator for ''F''-testing]

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:F-distribution}}
[[Category:Continuous distributions]]
[[Category:Analysis of variance]]