Editing Gumbel distribution (section)

==Properties==
The mode is μ, while the median is <math>\mu-\beta \ln\left(\ln 2\right),</math> and the mean is given by
:<math>\operatorname{E}(X)=\mu+\gamma\beta</math>,
where <math> \gamma </math> is the [[Euler–Mascheroni constant]].

The standard deviation <math> \sigma </math> is <math>\beta \pi/\sqrt{6}</math> hence <math>\beta = \sigma \sqrt{6} / \pi \approx 0.78 \sigma. </math> <ref name = "Oosterbaan" />

At the mode, where <math> x = \mu </math>, the value of <math>F(x;\mu,\beta)</math> becomes <math> e^{-1} \approx 0.37 </math>, irrespective of the value of <math> \beta. </math>

If <math>G_1,...,G_k</math> are iid Gumbel random variables with parameters <math>(\mu,\beta)</math> then <math>\max\{G_1,...,G_k\}</math> is also a Gumbel random variable with parameters <math>(\mu+\beta\ln k, \beta)</math>. 

If <math>G_1, G_2,...</math> are iid random variables such that <math>\max\{G_1,...,G_k\}-\beta\ln k </math> has the same distribution as <math>G_1</math> for all natural numbers <math> k </math>, then <math>G_1</math> is necessarily Gumbel distributed with scale parameter <math>\beta</math> (actually it suffices to consider just two distinct values of k>1 which are coprime).