The F-distribution

The F-distribution arises in statistics. If \(Y_1\) and \(Y_2\) are chi-squared deviates with \(\nu_1\) and \(\nu_2\) degrees of freedom then the ratio,

\[X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }\]

has an F-distribution \(F(x;\nu_1,\nu_2)\).

gsl_ran_fdist(nu1, nu2)

This function returns a random variate from the F-distribution with degrees of freedom nu1 and nu2. The distribution function is,

\[p(x) dx = { \Gamma((\nu_1 + \nu_2)/2) \over \Gamma(\nu_1/2) \Gamma(\nu_2/2) } \nu_1^{\nu_1/2} \nu_2^{\nu_2/2} x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}\]

for \(x \geq 0\).

gsl_ran_fdist_pdf(x, nu1, nu2)

This function computes the probability density \(p(x)\) at \(x\) for an F-distribution with nu1 and nu2 degrees of freedom, using the formula given above.

gsl_cdf_fdist_P(x, nu1, nu2)
gsl_cdf_fdist_Q(x, nu1, nu2)
gsl_cdf_fdist_Pinv(P, nu1, nu2)
gsl_cdf_fdist_Qinv(Q, nu1, nu2)

These functions compute the cumulative distribution functions \(P(x), Q(x)\) and their inverses for the F-distribution with nu1 and nu2 degrees of freedom.