F-Distribution
A continuous statistical distribution which arises in the testing of whether two observed samples have the same variance. Let 
and 
be independent variates distributed as chi-squared with 
and 
degrees
of freedom.
Define a statistic 
as the ratio of the dispersions
of the two distributions
 |
(1)
|
This statistic then has an 
-distribution on
domain 
with probability function 
and cumulative distribution function 
given by
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
is the gamma
function, 
is the beta
function, 
is the regularized
beta function, and 
is a hypergeometric
function.
The 
-distribution is implemented in the Wolfram Language as FRatioDistribution[n,
m].
The mean, variance, skewness and kurtosis excess are
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
The probability that 
would be as large as it is if the first
distribution has a smaller variance than the second is denoted 
.
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 946-949, 1972.
David, F. N. "The Moments of the 
and 
Distributions."
Biometrika 36, 394-403, 1949.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219-223, 1992.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 117-118, 1992.
Weisstein, Eric W. "F-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/F-Distribution.html
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