Distribution Gallery: Add Gamma (#506)

* Distribution Gallery: Add Gamma

* Apply suggestions from code review

Co-authored-by: Osvaldo A Martin <[email protected]>

---------

Co-authored-by: Osvaldo A Martin <[email protected]>

docs/examples/gallery/gamma.md

@@ -0,0 +1,98 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Gamma Distribution
+
+The gamma distribution is a continuous probability distribution that describes the time until a specified number of events occur in a Poisson process, in which $\alpha$ events occur continuously and independently at a constant average rate $\beta$. The gamma distribution is characterized by two parameters: the shape parameter $\alpha$ and the rate parameter $\beta$.
+
+The gamma distribution is widely used in many settings, including modeling the time until a specified number of decays of a radioactive atom, the time until a specified number of seizures in patients with epilepsy, the time until a specified number of failures of a machine, and the time until a specified number of changes in the price of a stock. 
+
+The gamma distribution is used as a conjugate prior for the exponential distribution and for the precision of the normal distribution. It can also used as prior for scale parameters, in particular when there is interest in avoiding values close to zero or one wants to include more information than with other prior commonly used for scale parameter like the halfnormal or the exponential.
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Gamma Distribution PDF
+---
+
+import arviz as az
+from preliz import Gamma
+az.style.use('arviz-doc')
+alphas = [1, 3, 7.5]
+betas = [0.5, 1., 1.]
+
+for alpha, beta in zip(alphas, betas):
+    Gamma(alpha, beta).plot_pdf()
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Gamma Distribution CDF
+---
+
+for alpha, beta in zip(alphas, betas):
+    Gamma(alpha, beta).plot_cdf()
+```
+
+## Key properties and parameters:
+
+```{eval-rst}
+
+========  ==========================================
+Support   :math:`x \in (0, \infty)`
+Mean      :math:`\frac{\alpha}{\beta}`
+Variance  :math:`\frac{\alpha}{\beta^2}`
+========  ==========================================
+```
+
+**Probability Density Function (PDF):**
+
+$$
+f(x|\alpha, \beta) = \frac{\beta^\alpha x^{\alpha - 1} e^{-\beta x}}{\Gamma(\alpha)}
+$$
+
+where $\Gamma$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function).
+
+**Cumulative Distribution Function (CDF):**
+
+$$
+F(x|\alpha, \beta) = \frac{1}{\Gamma(\alpha)} \gamma(\alpha, \beta x)
+$$
+
+where $\gamma$ is the [lower incomplete gamma function](https://en.wikipedia.org/wiki/Incomplete_gamma_function).
+
+```{seealso}
+:class: seealso
+
+**Common Alternatives:**
+
+- [Exponential Distribution](exponential.md) - A special case of the Gamma distribution with the shape parameter $\alpha = 1$.
+- [Chi-Squared Distribution](chisquared.md) - A special case of the Gamma distribution with the shape parameter $\alpha = \nu/2$ and the scale parameter $\beta = 1/2$.
+
+**Related Distributions:**
+
+- [Inverse Gamma Distribution](inverse-gamma.md) - The reciprocal of a gamma-distributed random variable.
+- [Poisson Distribution](poisson.md) - While the Gamma distribution models the time until a specified number of events occur in a Poisson process, the Poisson distribution models the number of events in fixed intervals of time or space.
+- [Negative Binomial Distribution](negative-binomial.md) - The discrete counterpart of the Gamma distribution, modeling the number of trials needed to achieve a specified number of successes in a sequence of Bernoulli trials.
+```
+
+
+## References
+
+- [Wikipedia - Gamma distribution](https://en.wikipedia.org/wiki/Gamma_distribution)

docs/gallery_content.rst

@@ -91,7 +91,7 @@ Continuous Distributions
       Exponential
 
    .. grid-item-card::
-      :link: ./api_reference.html#preliz.distributions.gamma.Gamma
+      :link: ./examples/gallery/gamma.html
       :text-align: center
       :shadow: none
       :class-card: example-gallery

docs/index.rst

@@ -1,4 +1,4 @@
-``````````````A tool-box for prior elicitation
+A tool-box for prior elicitation
 ===================================================
 
 |PyPI version|