Particle horizon

2014-01-28T18:14:36Z

205.175.116.124: /* Horizon problem */ much less than

{{Distinguish2|[[F-statistics]] as used in [[population genetics]]}}

{{Probability distribution |
name =Fisher-Snedecor|
type =density|
pdf_image =[[Image:F distributionPDF.png|325px]]|
cdf_image =[[Image:F distributionCDF.png|325px]]|
parameters =''d''1, ''d''2 > 0 deg. of freedom|
support = ''x'' ∈ [0, +∞)|
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> for ''d''2 > 2|
median =|
mode =<math>\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!</math> for ''d''1 > 2|
variance =<math>\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!</math> for ''d''2 > 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> for ''d''2 > 6|
kurtosis =''see text''|
entropy =|
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''' is a [[Continuous probability distribution|continuous]] [[probability distribution]].<ref name=johnson>{{cite book | last = Johnson
| first = Norman Lloyd
| coauthors = Samuel Kotz, 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
| coauthors = Franklin A. Graybill, Duane C. Boes
| title = Introduction to the Theory of Statistics (Third Edition, p. 246-249)
| publisher = McGraw-Hill
| year = 1974
| isbn = 0-07-042864-6}}</ref> It is also known as '''Snedecor's F distribution''' or the '''Fisher-Snedecor distribution''' (after [[Ronald Fisher|R.A. Fisher]] and [[George W. Snedecor]]). The F-distribution arises frequently as the [[null distribution]] of a [[test statistic]], most notably in the [[analysis of variance]]; see [[F-test]].

==Definition==
If a [[random variable]] ''X'' has an F-distribution with parameters ''d''1 and ''d''2, we write ''X'' ~ F(''d''1, ''d''2). Then the [[probability density function]] 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''1 and ''d''2 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''1, ''d''2) are given in the sidebox; for ''d''2 > 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''1, ''d''2) distribution exists and is finite only when 2''k'' < ''d''2 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''1 and ''d''2 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''1 and ''U''2 have [[chi-squared distribution]]s with ''d''1 and ''d''2 [[Degrees of freedom (statistics)|degrees of freedom]] respectively, and
*''U''1 and ''U''2 are [[statistical independence|independent]].

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

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

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

where ''s''12 and ''s''22 are the sums of squares ''S''12 and ''S''22 from two normal processes with variances σ12 and σ22 divided by the corresponding number of χ2 degrees of freedom, ''d''1 and ''d''2 respectively.(see [[Talk:F-distribution#Inconsistent.2C_or_at_least_confusing.2C_representation_in_terms_of_normal_variables|discussion]])

In a Frequentist context, a scaled F-distribution therefore gives the probability ''p''(''s''12/''s''22 | σ12, σ22), with the F distribution itself, without any scaling, applying where σ12 is being taken equal to σ22. This is the context in which the F-distribution most generally appears in [[F-test]]s: 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 σ12 and σ22.<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''(σ22/σ12|''s''12, ''s''22), where now the observed sums ''s''12 and ''s''22 are what are taken as known.

== Generalization ==
A generalization of the (central) F-distribution is the [[noncentral F-distribution]].

== Related distributions and properties ==
*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 \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''1, ''d''2), 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''1, ''d''2) 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''1, ''d''2) 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''1, ''d''2) then ''X''−1 ~ F(''d''2, ''d''1).
*If ''X'' ~ [[Student's t-distribution|t(''n'')]] 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> \tfrac{|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]] simplifies to the F-distribution if λ = 0.
*The doubly [[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''1, ''d''2) and <math>\operatorname{Q}_Y(1-p)</math> is the quantile 1−''p'' for ''Y'' ~ F(''d''2, ''d''1), then
::<math>\operatorname{Q}_X(p)=\frac{1}{\operatorname{Q}_Y(1-p)}</math>.

== See also ==
{{Colbegin}}
* [[Chi-squared distribution]]
* [[Chow test]]
* [[Gamma distribution]]
* [[Hotelling's T-squared distribution]]
* [[Student's t-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}}
{{Common univariate probability distributions}}

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

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