# Gumbel distributions

The Gumbel type I largest (resp. smallest) values distribution is usually referred as the extreme Value Type I maximum (resp. minimum) distribution. It is defined on the range $[-\infty, \infty]$, using the following characteristics.

• type I largest values
$\begin{matrix} &f_x(x) = \alpha\exp\left[-\alpha(x-u)\right] \cdot\exp\left[-\exp\left[-\alpha(x-u)\right]\right] \\ &F_x(x) = \exp\left[\exp\left[-\alpha(x-u)\right]\right] \\ \end{matrix}$

where $\alpha$ and $u$ are defined as:

$\begin{matrix} & \alpha = \frac{\pi}{\sigma\sqrt{6}} \qquad \alpha \ge 0 \\ &u = \mu - \frac{1}{\sqrt{3}\alpha} \end{matrix}$
• type I smallest values
$\begin{matrix} &f_x(x) = \alpha\exp\left[\alpha(x-u)\right] \cdot\exp\left[-\exp\left[\alpha(x-u)\right]\right] \\ &F_x(x) = 1-\exp[-\left[\exp\left[\alpha(x-u)\right]\right]] \\ \end{matrix}$

where $\alpha$ and $u$ are defined as:

$\begin{matrix} & \alpha = \frac{\pi}{\sigma\sqrt{6}} \qquad \alpha \ge 0 \\ &u = \mu - \frac{1}{\sqrt{3}\alpha} \end{matrix}$