F-distribution
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Probability density function None uploaded yet. | |
Cumulative distribution function None uploaded yet. | |
Parameters | deg. of freedom |
Support | |
cdf | |
Mean | for |
Median | |
Mode | for |
Variance | for |
Skewness | for |
Kurtosis | |
Entropy | |
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:
where
- U_{1} and U_{2} have chi-square distributions with d_{1} and d_{2} degrees of freedom respectively, and
- U_{1} and U_{2} 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(d_{1}, d_{2}) distributed random variable is given by
for real x ≥ 0, where d_{1} and d_{2} are positive integers, and B is the beta function.
The cumulative distribution function is
where I is the regularized incomplete beta function.
Generalization
A generalization of the (central) F-distribution is the noncentral F-distribution.
Related distributions
- is a chi-square distribution as for .
See also
References & Bibliography
Key texts
Books
Papers
Additional material
Books
Papers
External links
- 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-distributionde:F-Verteilung