The Gumbel distribution is a continuous probability distribution used to model the maximum (or minimum) of a sample. It is one of the three extreme-value distributions and describes the limiting distribution of the maximum of many independent light-tailed variables.
A random variable \(X\) follows a Gumbel distribution with location parameter \(\mu \in \mathbb{R}\) and scale parameter \(\beta > 0\) if
\[ X \sim \text{Gumbel}(\mu, \beta). \]
Writing \(z = \dfrac{x - \mu}{\beta}\), the probability density function is
\[ f(x) = \frac{1}{\beta} \exp\!\left(-\left(z + e^{-z}\right)\right), \quad x \in \mathbb{R}. \]
Interpretation: The density is right-skewed, with a heavier upper tail reflecting the occasional very large maximum.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.stats import gumbel_r
sns.set_style('whitegrid')
sns.set_palette('Set2')
x = np.linspace(-4, 8, 500)
fig, ax = plt.subplots(figsize=(7, 4))
for mu, beta in ((0, 1), (1, 2), (2, 1)):
ax.plot(x, gumbel_r.pdf(x, loc=mu, scale=beta),
lw=2, label=rf'$\mu={mu},\ \beta={beta}$')
ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
ax.set_title('Gumbel density')
ax.legend()
plt.show()
The cumulative distribution function is
\[ F(x) = \exp\!\left(-e^{-(x - \mu)/\beta}\right). \]
For \(X \sim \text{Gumbel}(\mu, \beta)\),
\[ \mathbb{E}[X] = \mu + \beta\gamma \quad \& \quad \operatorname{Var}(X) = \frac{\pi^2}{6}\beta^2, \]
where \(\gamma \approx 0.5772\) is the Euler–Mascheroni constant.
Proof. With \(z = (x - \mu)/\beta\) and the substitution \(u = e^{-z}\), the mean reduces to \(\mu + \beta\int_0^\infty (-\ln u) e^{-u} \, du = \mu + \beta\gamma\), using \(\int_0^\infty (-\ln u) e^{-u}\,du = \gamma\). The variance follows from the second logarithmic moment of the exponential.
The moment generating function (MGF) of \(X \sim \text{Gumbel}(\mu, \beta)\) is
\[ M_X(t) = e^{\mu t}\,\Gamma(1 - \beta t), \quad t < \frac{1}{\beta}, \]
where \(\Gamma\) is the gamma function.
Annual maximum river levels and rainfall extremes are modelled with the Gumbel distribution to estimate flood return periods.
Peak loads, maximum wind speeds, and material strength extremes are analysed using Gumbel models for design safety margins.
The Gumbel distribution of utility shocks gives rise to the logit model used throughout discrete-choice and machine-learning classification.