F-distribution

Not to be confused with F-statistics as used in population genetics.

Fisher-Snedecor

Probability density function
Cumulative distribution function
Parameters	$d_1>0,\ d_2>0$ deg. of freedom
Support	$x \in [0, +\infty)\!$
PDF	$\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!$
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 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	$\frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}\!$ for d2 > 6
Ex. kurtosis	see text
MGF	does not exist, raw moments defined in text and in
CF	see text

In probability theory and statistics, the F-distribution is a continuous probability distribution. 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, 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 $X\sim\operatorname{F}(d_1,d_2)$. Then the probability density function for X is given by

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

for real $x \ge 0$. Here 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

$F(x; d_1,d_2)=I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2) ,$

where I is the regularized incomplete beta function.

The expectation, variance, and other details about the $\operatorname{F}(d_1,d_2)$ 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 $\operatorname{F}(d_1,d_2)$ distribution exists and is finite only when 2k < d₂ and it is equal to :$\mu _{X}\left( k\right) =\left( \frac{d_{2}}{d_{1}}\right) ^{k}\frac{\Gamma \left( d_{1}/2+k\right) }{\Gamma \left( d_{1}/2\right) }\frac{\Gamma \left( d_{2}/2-k\right) }{\Gamma \left( d_{2}/2\right) }$

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

$\varphi^F_{d_1, d_2}(s) = \frac{\Gamma((d_1+d_2)/2)}{\Gamma(d_2/2)} U(d_1/2,1-d_2/2,-d_2 /d_1 \imath s)$

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:^{[citation needed]}

$\frac{U_1/d_1}{U_2/d_2}$

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 instance in the analysis of variance, independence of U₁ and U₂ might be demonstrated by applying Cochran's theorem.

Generalization

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

Related distributions and properties

• If $X \sim \chi^2_{\nu_1}$ and $Y \sim \chi^2_{\nu_2}$, and are independent, then $\frac{X / \nu_1}{Y / \nu_2} \sim \mathrm{F}(\nu_1, \nu_2)$
• If $X \sim \operatorname{Beta}(n/2,m/2)$ (Beta-distribution) then $\frac{m X}{n\left(1-X\right)} \sim \operatorname{F}(n,m)$
• If X∼F(ν₁,ν₂) then $Y = \lim_{\nu_2 \to \infty} \nu_1 X$ has the chi-squared distribution $\chi^2_{\nu_{1}}$
• $\operatorname{F}(\nu_1,\nu_2)$ is equivalent to the scaled Hotelling's T-squared distribution $\frac{\nu_2}{\nu_1(\nu_1+\nu_2-1)}\operatorname{T}^2(\nu_1,\nu_1+\nu_2-1)$.
• If $X \sim \operatorname{F}(\nu_1,\nu_2),$ then $\frac{1}{X} \sim F(\nu_2,\nu_1)$.
• If $X \sim \mathrm{t}(m)\,$ (Student's t-distribution) then $X^{2} \sim \operatorname{F}(\nu_1 = 1, \nu_2 = m)$.
• If $X \sim \mathrm{t}(n)\,$ (Student's t-distribution) then $X^{-2} \sim \operatorname{F}(\nu_1 = n, \nu_2 = 1)$.
• F-distribution is a special case of type 6 Pearson distribution
• If $X \sim \operatorname{F}(\nu_1,\nu_2)$ and $Y=\frac{\nu_1 X/\nu_2}{1+\nu_1 X/\nu_2}$ then $Y \sim \operatorname{Beta}(\nu_1/2,\nu_2/2)$ has a Beta-distribution.

• If X and Y are independent, with $X \sim \operatorname{Laplace}(\mu,b)$ and $Y \sim \operatorname{Laplace}(\mu,b)$ (Laplace distribution) then

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

• If $X \sim \operatorname{F}(n,m)$ 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 λ₁ = λ₂ = 0

• If $\operatorname{Q}_X(p)$ is the quantile p for $X\sim \operatorname{F}(\nu_1,\nu_2)$ and $\operatorname{Q}_Y(1-p)$ is the quantile 1 − p for $Y\sim \operatorname{F}(\nu_2,\nu_1)$, then

$\operatorname{Q}_X(p)=1/\operatorname{Q}_Y(1-p)$.

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