F-distribution
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Template:Probability distribution 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 Ronald Fisher and George W. Snedecor).
A random variate of the F-distribution arises as the ratio of two chi-squared variates:
- <math>\frac{U_1/d_1}{U_2/d_2}<math>
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 probability density function of an F(d1, d2) distributed random variable is given by
- <math> 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} <math>
for real x ≥ 0, where d1 and d2 are positive integers, and B is the beta function.
The cumulative distribution function is
- <math> G(x) = I_{\frac{d_1 x}{d_1 x + d_2}}(d_1/2, d_2/2) <math>
where I is the regularized incomplete beta function.
Generalization
A generalization of the (central) F-distribution is the noncentral F-distribution.
Related distributions
- <math>Y \sim \chi^2<math> is a chi-square distribution as <math>Y = \lim_{\nu_2 \to \infty} \nu_1 X<math> for <math>X \sim \mathrm{F}(\nu_1, \nu_2)<math>.
External links
- Table of critical values of the F-distribution (http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm)
- Online significance testing with the F-distribution (http://home.clara.net/sisa/signhlp.htm)
- Distribution Calculator (http://www.vias.org/simulations/simusoft_distcalc.html) Calculates probabilities and critical values for normal, t-, chi2- and F-distributionde:F-Verteilung