F distribution

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F distribution

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The F distribution, also known as Fisher's F or Fisher-Snedecor's distribution, is obtained as the ratio of two independent Chi-square distributions with n and m degrees of freedom. More specifically:

The probability density function (pdf) and higher moments are quite involved and details be found on the Mathworld and Wikipedia sites. The pdf can be written in various forms, but to keep the expression as simple as possible we will let a=n/2, b=m/2 and c=(m+n)/2. The pdf is then:

where B(a,b) is the Beta function. Because Chi-square distributions are themselves obtainable as the sum of squared Normal variates, the F-distribution is particularly useful for comparing rations of variances of Normal variates, which is major feature of Analysis of Variance (ANOVA) and regression analysis. Plots of the F-distribution for a range of values for n, and fixed m are provided in the diagram below.

Sample F-distribution graphs, for n=2,3,4 and m=5

fmulti

For any given n, the F distribution tends to a fixed form as m→∞. This form is the numerator of the ratio above, i.e. an adjusted Chi-square distribution. This is illustrated below, where n has been taken as fixed, at 4, and m increased to the limit.

Sample F-distribution graphs, for n=4 and m=1,2,10 and

fdistlimit

Key measures for the F distribution are provided in the table below:

Item

Value

Mean, μ1

m/(m-2), m>2

Variance, μ2

Skewness

see Mathworld

Kurtosis

see Mathworld

MGF

does not exist

Distribution tables are provided in the Resources topic, Distribution tables section

References

[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, II. Houghton Mifflin/J Wiley & Sons, New York

Mathworld: Weisstein E W: F-distribution: http://mathworld.wolfram.com/F-Distribution.html

Wikipedia: F distribution: http://en.wikipedia.org/wiki/F-distribution