Distributions Gallery: Add Gumbel

docs/examples/gallery/gumbel.md

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Gumbel Distribution
+
+The Gumbel distribution, also known as the type-I Generalized Extreme Value (GEV) distribution, is a continuous probability distribution that describes the distribution of the maximum (or minimum) of a number of samples of various different random variables, particularly the exponential and normal distributions. It is characterized by two parameters: the location parameter $\mu$ and the scale parameter $\beta$.
+
+The Gumbel distribution is commonly used in the fields of hydrology, meteorology, and environmental science to model extreme events such as floods, earthquakes, and wind speeds.
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Gumbel Distribution PDF
+---
+
+
+from preliz import Gumbel, style
+style.use('preliz-doc')
+mus = [0., 4., -1.]
+betas = [1., 2., 4.]
+for mu, beta in zip(mus, betas):
+    Gumbel(mu, beta).plot_pdf(support=(-10,20))
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Gumbel Distribution CDF
+---
+
+for mu, beta in zip(mus, betas):
+    Gumbel(mu, beta).plot_cdf(support=(-10,20))
+```
+
+## Key properties and parameters:
+
+```{eval-rst}
+
+========  ==========================================
+Support   :math:`x \in (-\infty, \infty)`
+Mean      :math:`\mu + \beta \gamma`, where :math:`\gamma` is the `Euler-Mascheroni constant <https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant>`_
+Variance  :math:`\frac{\pi^2}{6} \beta^2`
+========  ==========================================
+```
+
+**Probability Density Function (PDF):**
+
+$$
+f(x|\mu, \beta) = \frac{1}{\beta} e^{-(z + e^{-z})}
+$$
+
+where $z = \frac{x - \mu}{\beta}$.
+
+**Cumulative Distribution Function (CDF):**
+
+$$
+F(x|\mu, \beta) = e^{-e^{-z}}
+$$
+
+where $z = \frac{x - \mu}{\beta}$.
+
+```{seealso}
+:class: seealso
+
+**Related Distributions:**
+
+- [Weibull Distribution](weibull) - If a random variable follows a Weibull distribution, the logarithm of the random variable follows a Gumbel distribution.
+- [Exponential Distribution](exponential) - The maximum of a number of exponentially distributed random variables follows a Gumbel distribution and the exponential distribution is a special case of the Weibull distribution.
+```
+
+
+## References
+
+- [Wikipedia - Gumbel Distribution](https://en.wikipedia.org/wiki/Gumbel_distribution)

docs/gallery_content.rst

@@ -103,7 +103,7 @@ Continuous Distributions
       Gamma
 
    .. grid-item-card::
-      :link: ./api_reference.html#preliz.distributions.gumbel.Gumbel
+      :link: ./examples/gallery/gumbel.html
       :text-align: center
       :shadow: none
       :class-card: example-gallery