Gumbel (max.) Distribution
Gumbel (max.) Distribution#
The Gumbel (max.) distribution is a two-parameter continuous probability distribution. The table below summarizes some important aspects of the distribution.
Notation |
\(X \sim \mathrm{Gumbel}(\mu, \beta)\) |
Parameters |
\(\mu \in \mathbb{R}\) (location parameter) |
\(\beta > 0\) (scale parameter) |
|
\(\mathcal{D}_X = (-\infty, \infty)\) |
|
\(f_X (x; \mu, \beta) = \frac{1}{\beta} \exp{- \left[ \frac{x - \mu}{\beta} + \exp{-\left(\frac{x - \mu}{\beta} \right)} \right]}\) |
|
\(F_X (x; \mu, \beta) = \exp{-\left[ \exp{- \left(\frac{x - \mu}{\beta} \right)} \right]}\) |
|
\(F^{-1}_X (x; \mu, \beta) = \mu + \beta \ln{(\ln{x})}\) |
The plots of probability density functions (PDFs), sample histogram (of \(5'000\) points), cumulative distribution functions (CDFs), and inverse cumulative distribution functions (ICDFs) for different parameter values are shown below.
