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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}}
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 d2 > 2
Mode \frac{d_1-2}{d_1}\;\frac{d_2}{d_2+2}\! for d1 > 2
Variance \frac{2\,d_2^2\,(d_1+d_2-2)}{d_1 (d_2-2)^2 (d_2-4)}\! for d2 > 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 d2 > 6
Excess kurtosis see text
Moment-generating function (mgf) does not exist, raw moments defined elsewhere[1]
Characteristic function defined elsewhere[1]

In probability theory and statistics, the F-distribution is a continuous probability distribution.[1][2][3] 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.


[edit] Characterization

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



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}}

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(d1,d2) are given in the sidebox; for d2 > 8, the kurtosis is


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

[edit] Generalization

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

[edit] Related distributions and properties

[edit] References

  1. ^ 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 .
  2. ^ NIST (2006). Engineering Statistics Handbook - F Distribution
  3. ^ 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. 

[edit] External links

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