Distribution Gallery: Add HalfCauchy (#538)

docs/examples/gallery/halfcauchy.md

@@ -0,0 +1,81 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Half-Cauchy Distribution
+
+<audio controls> <source src="../../_static/halfcauchy.mp3" type="audio/mpeg"> This browser cannot play the pronunciation audio file for this distribution. </audio>
+
+The Half-Cauchy distribution is a continuous probability distribution that is derived from the Cauchy distribution but is restricted to only positive values. It is characterized by a single scale parameter ($\beta$), which determines the width of the distribution. Similar to the Cauchy distribution, the Half-Cauchy distribution has undefined mean and variance, making it an example of a "pathological" distribution with heavy tails.
+
+In Bayesian statistics, the Half-Cauchy distribution is often used as a prior for scale parameters.
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb: image
+---
+from preliz import HalfCauchy, style
+style.use('preliz-doc')
+betas = [0.5, 1, 2]
+for beta in betas:
+    HalfCauchy(beta).plot_pdf(support=(0, 5))
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell} 
+---
+tags: [remove-input]
+mystnb: image
+---
+for beta in betas:
+    HalfCauchy(beta).plot_cdf(support=(0, 5))
+```
+
+## Key properties and parameters:
+
+```{eval-rst}
+========  ==========================================
+Support   :math:`x \in [0, \infty)`
+Mean      undefined
+Variance  undefined
+========  ==========================================
+```
+
+**Probability Density Function (PDF):**
+
+$$ 
+f(x|\beta) = \frac{2}{\pi \beta \left[1 + \left(\frac{x}{\beta}\right)^2\right]}
+$$
+
+**Cumulative Distribution Function (CDF):**
+
+$$ 
+F(x|\beta) = \frac{2}{\pi} \arctan\left(\frac{x}{\beta}\right)
+$$
+
+```{seealso} 
+:class: seealso
+
+**Common Alternatives:**
+
+- [Half-Student's t](half_students_t.md) - The Half-Cauchy distribution is a special case of the Half-Student's t-distribution with $\nu=1$.
+- [Half-Normal](half_normal.md) - A distribution that considers only the positive half of the normal distribution.
+
+**Related Distributions:**
+
+- [Cauchy](cauchy.md) - The Cauchy distribution is the parent distribution from which the Half-Cauchy is derived.
+ ```
+
+## References
+
+- [Wikipedia - Cauchy](https://en.wikipedia.org/wiki/Cauchy_distribution)

docs/gallery_content.rst

@@ -115,7 +115,7 @@ Continuous Distributions
       Gumbel
 
    .. grid-item-card::
-      :link: ./api_reference.html#preliz.distributions.halfcauchy.HalfCauchy
+      :link: ./examples/gallery/halfcauchy.html
       :text-align: center
       :shadow: none
       :class-card: example-gallery