extreme value: Density of the Extreme Value Distribution of a Minimum.
Description
Density function of the extreme value distribution of a minimum
with location \(\alpha\) and scale \(\beta\)
and the density of the standardized version (with zero mean and unit variance).
Extreme value distribution of a minimum with the location \(\alpha\)
and the scale \(\beta\) has a density
$$f(x) = \frac{1}{\beta}\exp\left[\frac{x-\alpha}{\beta}-\exp\left(\frac{x-\alpha}{\beta}\right)\right]$$
the mean equal to \(\alpha - \beta\;e\), where \(e\) is approximately
\(0.5772\) and the variance equal to \(\beta^2\frac{\pi}{6}\).
Its standardized version is obtained with \(\alpha = \frac{\sqrt{6}}{\pi}\;e\)
and \(\beta = \frac{\sqrt{6}}{\pi}\)