F-distribution

From Wikipedia, the free encyclopedia

(Redirected from Snedecor's F distribution)

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)	see text for raw moments
Characteristic function

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).

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 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.

The expectation, variance, and skewness are given in the sidebox; for $d 2 > 8$ , the kurtosis is

$\frac{12(20d_2-8d_2^2+d_2^3+44d_1-32d_1d_2+5d_2^2d_1-22d_1^2+5d_2d_1^2-16)}{d_1(d_2-6)(d_2-8)(d_1+d_2-2)}.$

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

$g(x) = \frac{1}{\mathrm{B}(d_1/2, d_2/2)} \; \left(\frac{d_1\,x}{d_1\,x + d_2}\right)^{d_1/2} \; \left(1-\frac{d_1\,x}{d_1\,x + d_2}\right)^{d_2/2} \; x^{-1}$

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

The cumulative distribution function is

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

where I is the regularized incomplete beta function.

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

$Y \sim \chi^2$ has a chi-square distribution if $Y = \lim_{\nu_2 \to \infty} \nu_1 X$ for $X \sim \mathrm{F}(\nu_1, \nu_2)$ .
$F (ν 1,ν 2)$ is equivalent to the scaled Hotelling's T-square distribution $(ν 1 (ν 1 + ν 2 - 1) / ν 2) T 2 (ν 1,ν 1 + ν 2 - 1)$ .
One interesting property is that if $X \sim F(\nu_1,\nu_2),\ \frac{1}{X} \sim F(\nu_2,\nu_1)$ .

Table of critical values of the F-distribution
Online significance testing with the F-distribution
Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
Cumulative density function (CDF) calculator for the Fisher F-distribution
Probability density function (PDF) calculator for the Fisher F-distribution

	Probability distributions [ view • talk • edit ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Generalized Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	bimodal • Cantor • conditional • equilibrium • exponential family • Infinite divisibility (probability) • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular • unimodal

Retrieved from "http://en.wikipedia.org/wiki/F-distribution"

Categories: Continuous distributions | Analysis of variance

Searching reference...

Reference

Sources

F-distribution

From Wikipedia, the free encyclopedia

Top Matching Results

Sponsored Links
This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Searching reference...

Reference

Sources

F-distribution

From Wikipedia, the free encyclopedia

Top Matching Results

Sponsored Links This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Sponsored Links
This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.