F-distribution

Fisher-Snedecor

Probability density function
Cumulative distribution function
Parameters	d1, d2 > 0 deg. of freedom
Support	x ∈ [0, +∞)
PDF	${\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\!$
CDF	$I_{\frac {d_{1}x}{d_{1}x+d_{2}}}\left({\tfrac {d_{1}}{2}},{\tfrac {d_{2}}{2}}\right)$
Mean	${\frac {d_{2}}{d_{2}-2}}\!$ for d2 > 2
Mode	${\frac {d_{1}-2}{d_{1}}}\;{\frac {d_{2}}{d_{2}+2}}$ for d1 > 2
Variance	${\frac {2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)}}\!$ for d2 > 4
Skewness	${\frac {(2d_{1}+d_{2}-2){\sqrt {8(d_{2}-4)}}}{(d_{2}-6){\sqrt {d_{1}(d_{1}+d_{2}-2)}}}}\!$ for d2 > 6
Excess kurtosis	see text
MGF	does not exist, raw moments defined in text and in
CF	see text

The F-distribution, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is, in probability theory and statistics, a continuous probability distribution. 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 d₁ and d₂, we write X ~ F(d₁, d₂). Then the probability density function (pdf) for X is given by

${\begin{aligned}f(x;d_{1},d_{2})&={\frac {\sqrt {\frac {(d_{1}\,x)^{d_{1}}\,\,d_{2}^{d_{2}}}{(d_{1}\,x+d_{2})^{d_{1}+d_{2}}}}}{x\,\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\\&={\frac {1}{\mathrm {B} \!\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\left({\frac {d_{1}}{d_{2}}}\right)^{\frac {d_{1}}{2}}x^{{\frac {d_{1}}{2}}-1}\left(1+{\frac {d_{1}}{d_{2}}}\,x\right)^{-{\frac {d_{1}+d_{2}}{2}}}\end{aligned}}$

for real x ≥ 0. Here $\mathrm {B}$ 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.

The cumulative distribution function is

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

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

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

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

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

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., ). The correct expression is

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

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

Characterization

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

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

where

• U₁ and U₂ have chi-squared distributions with d₁ and d₂ degrees of freedom respectively, and
• U₁ and U₂ are independent.

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

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

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.

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

Differential equation

The probability density function of the F-distribution is a solution of the following differential equation:

$\left\{{\begin{array}{l}2x\left(d_{1}x+d_{2}\right)f'(x)+\left(2d_{1}x+d_{2}d_{1}x-d_{2}d_{1}+2d_{2}\right)f(x)=0,\\[12pt]f(1)={\frac {d_{1}^{\frac {d_{1}}{2}}d_{2}^{\frac {d_{2}}{2}}\left(d_{1}+d_{2}\right){}^{{\frac {1}{2}}\left(-d_{1}-d_{2}\right)}}{B\left({\frac {d_{1}}{2}},{\frac {d_{2}}{2}}\right)}}\end{array}}\right\}$

Generalization

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

Related distributions and properties

• If $X\sim \chi _{d_{1}}^{2}$ and $Y\sim \chi _{d_{2}}^{2}$ are independent, then ${\frac {X/d_{1}}{Y/d_{2}}}\sim \mathrm {F} (d_{1},d_{2})$
• If $X\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)$ (Beta distribution) then ${\frac {d_{2}X}{d_{1}(1-X)}}\sim \operatorname {F} (d_{1},d_{2})$
• Equivalently, if X ~ F(d₁, d₂), then ${\frac {d_{1}X/d_{2}}{1+d_{1}X/d_{2}}}\sim \operatorname {Beta} (d_{1}/2,d_{2}/2)$.
• If X ~ F(d₁, d₂) then $Y=\lim _{d_{2}\to \infty }d_{1}X$ has the chi-squared distribution $\chi _{d_{1}}^{2}$
• F(d₁, d₂) is equivalent to the scaled Hotelling's T-squared distribution ${\frac {d_{2}}{d_{1}(d_{1}+d_{2}-1)}}\operatorname {T} ^{2}(d_{1},d_{1}+d_{2}-1)$.
• If X ~ F(d₁, d₂) then X⁻¹ ~ F(d₂, d₁).
• If X ~ t(n) then

$X^{2}\sim \operatorname {F} (1,n)$

$X^{-2}\sim \operatorname {F} (n,1)$

• F-distribution is a special case of type 6 Pearson distribution

• If X and Y are independent, with X, Y ~ Laplace(μ, b) then

${\tfrac {|X-\mu |}{|Y-\mu |}}\sim \operatorname {F} (2,2)$

• If X ~ F(n, m) then ${\tfrac {\log {X}}{2}}\sim \operatorname {FisherZ} (n,m)$ (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 $\lambda _{1}=\lambda _{2}=0$

• If $\operatorname {Q} _{X}(p)$ is the quantile p for X ~ F(d₁, d₂) and $\operatorname {Q} _{Y}(1-p)$ is the quantile 1−p for Y ~ F(d₂, d₁), then

$\operatorname {Q} _{X}(p)={\frac {1}{\operatorname {Q} _{Y}(1-p)}}$.

See also

• Chi-squared distribution
• Chow test
• Gamma distribution
• Hotelling's T-squared distribution
• Student's t-distribution
• Wilks' lambda distribution
• Wishart distribution

External links

• Table of critical values of the F-distribution
• Earliest Uses of Some of the Words of Mathematics: entry on F-distribution contains a brief history
• Free calculator for F-testing