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In statistics and probability, 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
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.
An F(d1, d2) random variable has the following properties:
- mode
- <math> \frac{d_1-2}{d_1}\cdot\frac{d_2}{d_2+2} <math> provided d1 > 2
- mean
- <math>\mu = \frac{d_2}{d_2-2}<math> provided d2 > 2
- variance
- <math>\sigma^2 = \frac{2 d_2^2 (d_1 + d_2 - 2)}{d_1 (d_2-2)^2 (d_2-4)} <math> provided d2 > 4
- skewness
- <math>\gamma_1 = \frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2-4)}}{(d_2-6) \sqrt{d_1 (d_1 + d_2 -2)}}<math> provided d2 > 6
Generalization
A generalization of the (central) F-distribution is the noncentral F-distribution.
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