F-distribution

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Fisher-Snedecor
Probability density function
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Cumulative distribution function
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Parameters deg. of freedom
Support
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean for
Median
Mode for
Variance for
Skewness
for
Excess kurtosis see text
Entropy
Moment-generating function (mgf) see text for raw moments
Characteristic function

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 R.A. Fisher and George W. Snedecor).

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

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 expectation, variance, and skewness are given in the sidebox; for , the kurtosis is

The probability density function of an F(d1, d2) distributed random variable is given by

for real x ≥ 0, where d1 and d2 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 and properties

  • has a chi-square distribution if for .
  • is equivalent to the scaled Hotelling's T-square distribution .
  • One interesting property is that if .

External links

de:F-Verteilung it:Variabile casuale F di Snedecor nl:F-verdeling su:Sebaran-F fi:F-jakauma




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