F-Distribution
Arises in the testing of whether two observed samples have the same Variance. Let 
and 
beindependent variates distributed as Chi-Squared with 
and 
Degrees ofFreedom. Define a statistic 
as the ratio of the dispersions of the two distributions
 | (1) |
-distribution with probability function and cumulative distribution |  |  | (2) |
|  | (3) | |
|  | (4) |
where
is the Gamma Function, 
is the Beta Function, and 
is the Regularized Beta Function. The Mean, Variance, Skewness and Kurtosis are | |  | (5) |
 | |  | (6) |
 | |  | (7) |
 | |  | |
 |  | (8) |
The probability that
would be as large as it is if the first distribution has a smaller variance than the second isdenoted 
.The noncentral -distribution is given by
 | |  | |
|  | (9) |
where
is the Gamma Function, 
is the Beta Function, and 
is an associated Laguerre Polynomial.See also Beta Function, Gamma Function, Regularized Beta Function, Snedecor's F-Distribution
References
Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 946-949, 1972.
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.
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