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Probability density function
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Cumulative distribution function
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Parameters d_1>0,\ d_2>0 deg. of freedom
Support x \in [0; +\infty)\!
pdf \frac{\sqrt{\frac{(d_1\,x)^{d_1}\,\,d_2^{d_2}}
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
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
mgf see text for raw moments
Char. func.

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:



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

 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.

Related distributions

External links

es:Distribución Fnl:F-verdeling

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