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	$x \in [0, %2B\infty)\!$
PDF	$\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x%2Bd_2)^{d_1%2Bd_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!$
CDF	$I_{\frac{d_1 x}{d_1 x %2B d_2}}(d_1/2, d_2/2)\!$
Mean	$\frac{d_2}{d_2-2}\!$ for $d_2 > 2$
Mode	$\frac{d_1-2}{d_1}\;\frac{d_2}{d_2%2B2}\!$ for $d_1 > 2$
Variance	$\frac{2\,d_2^2\,(d_1%2Bd_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!$ for $d_2 > 4$
Skewness	$\frac{(2 d_1 %2B d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 %2B d_2 -2)}}\!$ for $d_2 > 6$
Ex. kurtosis	see text
MGF	does not exist, raw moments defined in text and in ^[1]^[2]
CF	see text

In probability theory and statistics, the F-distribution is a continuous probability distribution.^[1]^[2]^[3]^[4] 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.

1 Definition
2 Characterization
3 Generalization
4 Related distributions and properties
5 References
6 External links

Definition

If a random variable $X$ has an F-distribution with parameters $d_1$ and $d_2$ , we write $X\sim\operatorname{F}(d_1,d_2)$ . Then the probability density function for $X$ is given by

$\begin{align} f(x; d_1,d_2) &= \frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x%2Bd_2)^{d_1%2Bd_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \\ &=\frac{1}{\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)} \left(\frac{d_1}{d_2}\right)^{\frac{d_1}{2}} x^{\frac{d_1}{2} - 1} \left(1%2B\frac{d_1}{d_2}\,x\right)^{-\frac{d_1%2Bd_2}{2}} \! \end{align}$

for real $x \ge 0$ . Here $\mathrm{B}$ is the beta function. In many applications, the parameters $d_1$ and $d_2$ are positive integers, but the distribution is well-defined for positive real values of these parameters.

The cumulative distribution function is

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

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 $d_2>8$ , the excess kurtosis is

$\gamma_2 = 12\frac{d_1(5d_2-22)(d_1%2Bd_2-2)%2B(d_2-4)(d_2-2)^2}{d_1(d_2-6)(d_2-8)(d_1%2Bd_2-2)}$ .

The k-th moment of an $\operatorname{F}(d_1,d_2)$ distribution exists and is finite only when $2k<d_2$ and it is equal to ^[5]: $\mu _{X}\left( k\right) =\left( \frac{d_{2}}{d_{1}}\right) ^{k}\frac{\Gamma \left( d_{1}/2%2Bk\right) }{\Gamma \left( d_{1}/2\right) }\frac{\Gamma \left( d_{2}/2-k\right) }{\Gamma \left( d_{2}/2\right) }$

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., ^[2]). The correct expression ^[6] is

$\varphi^F_{d_1, d_2}(s) = \frac{\Gamma((d_1%2Bd_2)/2)}{\Gamma(d_2/2)} U(d_1/2,1-d_2/2,-d_2 /d_1 \imath s)$

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:

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

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 $X \sim \mathrm{F}(\nu_1, \nu_2)$ 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 $\frac{\nu_2}{\nu_1(\nu_1%2B\nu_2-1)}\operatorname{T}^2(\nu_1,\nu_1%2B\nu_2-1)$ .
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 $Y=\frac{\nu_1 X/\nu_2}{1%2B\nu_1 X/\nu_2}$ 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 $\lambda = 0$
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 $X\sim \operatorname{F}(\nu_1,\nu_2)$ and $\operatorname{Q}_Y(1-p)$ is the quantile $1-p$ for $Y\sim \operatorname{F}(\nu_2,\nu_1)$ , then

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

References

^ ^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.
^ ^a ^b ^c Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 26", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 946, ISBN 978-0486612720, MR 0167642, http://www.math.sfu.ca/~cbm/aands/page_946.htm .
^ NIST (2006). Engineering Statistics Handbook - F Distribution
^ Mood, Alexander; Franklin A. Graybill, Duane C. Boes (1974). Introduction to the Theory of Statistics (Third Edition, p. 246-249). McGraw-Hill. ISBN 0-07-042864-6.
^ Taboga, Marco. "The F distribution". http://www.statlect.com/F_distribution.htm.
^ Phillips, P. C. B. (1982) "The true characteristic function of the F distribution," Biometrika, 69: 261-264 JSTOR 2335882

External links

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