## F-Distribution

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 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 is denoted .

The noncentral -distribution is given by

 (9)

where is the Gamma Function, is the Beta Function, and is an associated Laguerre Polynomial.

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.