# F-distribution

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.[2][3][4][5]

Parameters Probability density function Cumulative distribution function d1, d2 > 0 deg. of freedom x ∈ [0, +∞) ${\displaystyle {\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!}$ ${\displaystyle I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)}$ ${\displaystyle {\frac {d_{2}}{d_{2}-2}}\!}$ for d2 > 2 ${\displaystyle {\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}}$ for d1 > 2 ${\displaystyle {\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!}$ for d2 > 4 ${\displaystyle {\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!}$for d2 > 6 see text ${\displaystyle \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)+\!}$${\displaystyle \left(1-{\tfrac {d_{1}}{2}}\right)\psi \left(1+{\tfrac {d_{1}}{2}}\right)-\left(1+{\tfrac {d_{2}}{2}}\right)\psi \left(1+{\tfrac {d_{2}}{2}}\right)\!}$${\displaystyle +\left({\tfrac {d_{1}+d_{2}}{2}}\right)\psi \left({\tfrac {d_{1}+d_{2}}{2}}\right)+\ln {\frac {d_{1}}{d_{2}}}\!}$[1] does not exist, raw moments defined in text and in [2][3] see text

## DefinitionEdit

If a random variable X has an F-distribution with parameters d1 and d2, we write X ~ F(d1, d2). Then the probability density function (pdf) for X is given by

{\displaystyle {\begin{aligned}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{aligned}}}

for real x > 0. Here ${\displaystyle \mathrm {B} }$  is the beta function. In many applications, the parameters d1 and d2 are positive integers, but the distribution is well-defined for positive real values of these parameters.

${\displaystyle F(x;d_{1},d_{2})=I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right),}$

where I is the regularized incomplete beta function.

The expectation, variance, and other details about the F(d1, d2) are given in the sidebox; for d2 > 8, the excess kurtosis is

${\displaystyle \gamma _{2}=12{\frac {d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}}$ .

The k-th moment of an F(d1, d2) distribution exists and is finite only when 2k < d2 and it is equal to [6]

${\displaystyle \mu _{X}(k)=\left({\frac {d_{2}}{d_{1}}}\right)^{k}{\frac {\Gamma \left({\tfrac {d_{1}}{2}}+k\right)}{\Gamma \left({\tfrac {d_{1}}{2}}\right)}}{\frac {\Gamma \left({\tfrac {d_{2}}{2}}-k\right)}{\Gamma \left({\tfrac {d_{2}}{2}}\right)}}}$

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 is listed incorrectly in many standard references (e.g.,[3]). The correct expression [7] is

${\displaystyle \varphi _{d_{1},d_{2}}^{F}(s)={\frac {\Gamma ({\frac {d_{1}+d_{2}}{2}})}{\Gamma ({\tfrac {d_{2}}{2}})}}U\!\left({\frac {d_{1}}{2}},1-{\frac {d_{2}}{2}},-{\frac {d_{2}}{d_{1}}}\imath s\right)}$

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

## CharacterizationEdit

A random variate of the F-distribution with parameters d1 and d2 arises as the ratio of two appropriately scaled chi-squared variates:[8]

${\displaystyle X={\frac {U_{1}/d_{1}}{U_{2}/d_{2}}}}$

where

In instances where the F-distribution is used, for example in the analysis of variance, independence of U1 and U2 might be demonstrated by applying Cochran's theorem.

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

${\displaystyle X={\frac {s_{1}^{2}}{\sigma _{1}^{2}}}\;/\;{\frac {s_{2}^{2}}{\sigma _{2}^{2}}}}$

where s12 and s22 are the sums of squares S12 and S22 from two normal processes with variances σ12 and σ22 divided by the corresponding number of χ2 degrees of freedom, d1 and d2 respectively : ${\displaystyle s_{1}^{2}={\frac {S_{1}^{2}}{d_{1}}}}$  and ${\displaystyle s_{2}^{2}={\frac {S_{2}^{2}}{d_{2}}}}$ .[discuss][citation needed]

In a frequentist context, a scaled F-distribution therefore gives the probability p(s12/s22 | 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-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 probabilities of σ12 and σ22.[9] In this context, a scaled F-distribution thus gives the posterior probability p2212|s12, s22), where now the observed sums s12 and s22 are what are taken as known.

## Properties and related distributionsEdit

• If ${\displaystyle X\sim \chi _{d_{1}}^{2}}$  and ${\displaystyle Y\sim \chi _{d_{2}}^{2}}$  are independent, then ${\displaystyle {\frac {X/d_{1}}{Y/d_{2}}}\sim \mathrm {F} (d_{1},d_{2})}$
• If ${\displaystyle X_{k}\sim \Gamma (\alpha _{k},\beta _{k})\,}$  are independent, then ${\displaystyle {\frac {\alpha _{2}\beta _{1}X_{1}}{\alpha _{1}\beta _{2}X_{2}}}\sim \mathrm {F} (2\alpha _{1},2\alpha _{2})}$
• If ${\displaystyle X\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)}$  (Beta distribution) then ${\displaystyle {\frac {d_{2}X}{d_{1}(1-X)}}\sim \operatorname {F} (d_{1},d_{2})}$
• Equivalently, if X ~ F(d1, d2), then ${\displaystyle {\frac {d_{1}X/d_{2}}{1+d_{1}X/d_{2}}}\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)}$ .
• If X ~ F(d1, d2), then ${\displaystyle {\frac {d_{1}}{d_{2}}}X}$  has a beta prime distribution: ${\displaystyle {\frac {d_{1}}{d_{2}}}X\sim \operatorname {\beta ^{\prime }} ({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}})}$ .
• If X ~ F(d1, d2) then ${\displaystyle Y=\lim _{d_{2}\to \infty }d_{1}X}$  has the chi-squared distribution ${\displaystyle \chi _{d_{1}}^{2}}$
• F(d1, d2) is equivalent to the scaled Hotelling's T-squared distribution ${\displaystyle {\frac {d_{2}}{d_{1}(d_{1}+d_{2}-1)}}\operatorname {T} ^{2}(d_{1},d_{1}+d_{2}-1)}$ .
• If X ~ F(d1, d2) then X−1 ~ F(d2, d1).
• If X ~ t(n) -- Student's t-distribution—then:
${\displaystyle X^{2}\sim \operatorname {F} (1,n)}$
${\displaystyle X^{-2}\sim \operatorname {F} (n,1)}$
${\displaystyle {\tfrac {|X-\mu |}{|Y-\mu |}}\sim \operatorname {F} (2,2)}$
• If X ~ F(n, m) then ${\displaystyle {\tfrac {\log {X}}{2}}\sim \operatorname {FisherZ} (n,m)}$  (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 ${\displaystyle \lambda _{1}=\lambda _{2}=0}$
• If ${\displaystyle \operatorname {Q} _{X}(p)}$  is the quantile p for X ~ F(d1, d2) and ${\displaystyle \operatorname {Q} _{Y}(1-p)}$  is the quantile 1−p for Y ~ F(d2, d1), then
${\displaystyle \operatorname {Q} _{X}(p)={\frac {1}{\operatorname {Q} _{Y}(1-p)}}}$ .

## ReferencesEdit

1. ^ Lazo, A.V.; Rathie, P. (1978). "On the entropy of continuous probability distributions". IEEE Transactions on Information Theory. IEEE. 24 (1): 120–122.
2. ^ a b Johnson, Norman Lloyd; Samuel Kotz; N. Balakrishnan (1995). Continuous Univariate Distributions, Volume 2 (Second Edition, Section 27). Wiley. ISBN 0-471-58494-0.
3. ^ a b c Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 26". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 946. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
4. ^ NIST (2006). Engineering Statistics Handbook – F Distribution
5. ^ Mood, Alexander; Franklin A. Graybill; Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, pp. 246–249). McGraw-Hill. ISBN 0-07-042864-6.
6. ^ Taboga, Marco. "The F distribution".
7. ^ Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261–264 JSTOR 2335882
8. ^ M.H. DeGroot (1986), Probability and Statistics (2nd Ed), Addison-Wesley. ISBN 0-201-11366-X, p. 500
9. ^ G.E.P. Box and G.C. Tiao (1973), Bayesian Inference in Statistical Analysis, Addison-Wesley. p.110