Returns the PDF of the Gumbel distribution.
pgumbel(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)Vector of quantiles
The location parameter, \(mu\). This is not the mean of the Gumbel distribution.
The scale parameter \(sigma\).
Logical, whether lower, if TRUE or upper, if FALSE, tail probabilities should be returned.
Logical, if TRUE probabilities are given in their log.
A vector of probabilities
This code and the details of the help file were taken from the VGAM package.
The Gumbel distribution is a special case of the generalized extreme value
(GEV) distribution where the shape parameter \(\xi\) = 0. The latter has 3 parameters, so
the Gumbel distribution has two. The Gumbel distribution function is
$$G(y) = \exp \left( - \exp \left[ - \frac{y-\mu}{\sigma} \right]\right) $$
where \(-\infty<y<\infty\), \(-\infty<\mu<\infty\)and \(\sigma>0\).
Its mean is
$$\mu - \sigma * \gamma$$
and its variance is
$$\sigma^2 * \pi^2 / 6$$
where \(\gamma\) is Euler's constant (which can be obtained as -digamma(1)).