F-distribution: Information from Answers.com

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	$x \in [0, +\infty)\!$
Probability density function (pdf)	$\frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!$
Cumulative distribution function (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 $d 2 > 2$
Median
Mode	$\frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\!$ for $d 1 > 2$
Variance	$\frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\!$ for $d 2 > 4$
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 $d 2 > 6$
Excess kurtosis	see text
Entropy
Moment-generating function (mgf)	does not exist, raw moments defined elsewhere^[1]^[2]
Characteristic function	defined elsewhere^[1]^[2]

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, especially in likelihood-ratio tests, perhaps most notably in the analysis of variance; see F-test.

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

Characterization

A random variate of the F-distribution arises as the ratio of two chi-squared variates:

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

where

U₁ and U₂ have chi-square distributions with d₁ and d₂ degrees of freedom respectively, and

U₁ and U₂ are independent (see Cochran's theorem for an application).

The probability density function of an F(d₁, d₂) distributed random variable is given by

$f(x) = \frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}} {(d_1\,x+d_2)^{d_1+d_2}}}} {x\,\mathrm{B}\!\left(\frac{d_1}{2},\frac{d_2}{2}\right)}\!$

for real x ≥ 0, where d₁ and d₂ are positive integers, and B is the beta function.

The cumulative distribution function is $F(x)=I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2)$

where I is the regularized incomplete beta function.

The expectation, variance, and other details about the $F (d 1, d 2)$ are given in the sidebox; for $d 2 > 8$ , the kurtosis is

$\frac{20d_2-8d_2^2+d_2^3+44d_1-32d_1d_2+A}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)/12}$

where $A=5d_2^2d_1-22d_1^2+5d_2d_1^2-16.$

The F-distribution is a particular parameterisation of the beta prime distribution, which is also called the beta distribution of the second kind.

Generalization

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

Related distributions and properties

If $X \sim \mathrm{F}(\nu_1, \nu_2)$ then $Y = \lim_{\nu_2 \to \infty} \nu_1 X$ has the chi-square distribution $\chi^2_{\nu_{1}}$
$\operatorname{F}(\nu_1,\nu_2)$ is equivalent to the scaled Hotelling's T-square distribution $(\nu_1(\nu_1+\nu_2-1)/\nu_2)\operatorname{T}^2(\nu_1,\nu_1+\nu_2-1)$ .
If $X \sim \operatorname{F}(\nu_1,\nu_2),$ then $\frac{1}{X} \sim F(\nu_2,\nu_1)$ .
if $X \sim \mathrm{t}(\nu)\!$ has a Student's t-distribution then $X^2 \sim \operatorname{F}(\nu_1 = 1, \nu_2 = \nu)$ .
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 $\operatorname{Q}_X(p)$ 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 ^c 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, ISBN 0-486-61272-4 .
^ 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.

External links

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Beta · Irwin–Hall · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Beta prime · Bose–Einstein · Burr · chi-square · chi · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scaled inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · generalized normal · generalized hyperbolic · Gumbel · hyperbolic secant · Landau · Laplace · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · slash · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · mixture · Pearson · stable · Tweedie

v • d • e Some common univariate probability distributions

Continuous	beta • Cauchy • chi-square • exponential • F • gamma • Laplace • log-normal • normal • Pareto • Student's t • uniform • Weibull

Discrete	Bernoulli • binomial • discrete uniform • geometric • hypergeometric • negative binomial • Poisson

List of probability distributions

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F-distribution

Contents

Characterization

Generalization

Related distributions and properties

References

External links