Gumbel Distribution
- $\mu \in \mathbb{R}$,
- given
- $\beta > 0$,
- given
$\mathrm{Gu}(x; \mu, \beta)$ is a p.d.f. of gumbel distribution given $\mu$ and $\beta$ defined by
\[\begin{eqnarray} \mathrm{Gu}(x; \mu, \beta) & := & \frac{1}{\beta} \exp \left( - \left( z + e^{-z} \right) \right) \nonumber \\ z & := & \frac{ x - \mu }{ \beta } \nonumber \end{eqnarray}\]$\mathrm{Gu}(x; \mu, \beta)$ is a c.d.f. of gumbel distribution given $\mu$ and $\beta$ defined by
\[\begin{eqnarray} \mathrm{Gu}(x; \mu, \beta) & := & \exp \left( -e^{-z} \right) \nonumber \\ z & := & \frac{ x - \mu }{ \beta } \nonumber \end{eqnarray}\]