2.1.1.11. GumbelMax law

This law describes a Gumbel max distribution depending on the mode \(\mu\) and the scale \(\beta\), as

\[f(x) = z \times\frac{e^{-z}}{\beta}, \; \mbox{where} \; z = e^{\frac{-(x - \mu)}{\beta}}\]

The mean value of the Gumbel max law can then be computed as \({\rm mean} = \mu+\beta\gamma\), where \(\gamma\) is the Euler Mascheroni constant and its variance can be written as \(\sigma^2=\dfrac{\pi^{2}}{6}\beta^{2}\)

Figure 2.12 shows the PDF, CDF and inverse CDF generated for different sets of parameters.

Figure 2.12 Example of PDF, CDF and inverse CDF for GumbelMax distributions.