The Type-1 Gumbel Distribution

gsl_ran_gumbel1(a, b)

This function returns a random variate from the Type-1 Gumbel distribution. The Type-1 Gumbel distribution function is,

\[p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx\]

for \(-\infty < x < \infty\).

gsl_ran_gumbel1_pdf(x, a, b)

This function computes the probability density \(p(x)\) at \(x\) for a Type-1 Gumbel distribution with parameters a and b, using the formula given above.

gsl_cdf_gumbel1_P(x, a, b)
gsl_cdf_gumbel1_Q(x, a, b)
gsl_cdf_gumbel1_Pinv(P, a, b)
gsl_cdf_gumbel1_Qinv(Q, a, b)

These functions compute the cumulative distribution functions \(P(x), Q(x)\) and their inverses for the Type-1 Gumbel distribution with parameters a and b.