Distributions Gallery: Add BetaBinomial

docs/examples/gallery/betabinomial.md

@@ -0,0 +1,96 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Beta-Binomial Distribution
+
+<audio controls> <source src="../../_static/betabinomial.mp3" type="audio/mpeg"> This browser cannot play the pronunciation audio file for this distribution. </audio>
+
+The Beta-Binomial distribution is a discrete probability distribution derived from the Binomial distribution, with its probability of success in each trial governed by a Beta distribution. This distribution is characterized by three parameters:  $\alpha$, $\beta$, and $n$, where $\alpha$ and $\beta$ are the shape parameters of the Beta distribution and $n$ is the number of trials in the Binomial distribution.
+
+
+
+## Probability Mass Function (PMF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Beta-Binomial Distribution PMF
+---
+
+from preliz import BetaBinomial, style
+style.use('preliz-doc')
+alphas = [0.5, 1, 2.3]
+betas = [0.5, 1, 2]
+n = 10
+for a, b in zip(alphas, betas):
+    BetaBinomial(a, b, n).plot_pdf()
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Beta-Binomial Distribution CDF
+---
+
+for a, b in zip(alphas, betas):
+    BetaBinomial(a, b, n).plot_cdf()
+```
+
+## Key properties and parameters:
+
+```{eval-rst}
+========  =================================================================
+Support   :math:`x \in \{0, 1, \ldots, n\}`
+Mean      :math:`n \dfrac{\alpha}{\alpha + \beta}`
+Variance  :math:`\dfrac{n \alpha \beta (\alpha+\beta+n)}{(\alpha+\beta)^2 (\alpha+\beta+1)}`
+========  =================================================================
+```
+
+**Probability Mass Function (PMF):**
+
+$$
+f(x \mid \alpha, \beta, n) = \binom{n}{x} \frac{B(x + \alpha, n - x + \beta)}{B(\alpha, \beta)}
+$$
+
+where $B$ is the [Beta function](https://en.wikipedia.org/wiki/Beta_function) and $\binom{n}{x}$ is the [binomial coefficient](https://en.wikipedia.org/wiki/Binomial_coefficient).
+
+**Cumulative Distribution Function (CDF):**
+
+$$
+F(x \mid \alpha, \beta, n) = 
+\begin{cases} 
+0, & x < 0 \\ 
+\sum_{k=0}^{x} \binom{n}{k} \frac{B(k + \alpha, n - k + \beta)}{B(\alpha, \beta)}, & 0 \leq x < n \\ 
+1, & x \geq n 
+\end{cases}
+$$
+
+```{seealso}
+:class: seealso
+
+**Related Distributions:**
+
+- [Binomial](binomial.md) - The Beta-Binomial distribution is a compound distribution derived from the Binomial distribution.
+- [Beta](beta.md) - The Beta-Binomial distribution is a compound distribution parameterized by the Beta distribution.
+```
+
+## References
+
+- [Wikipedia - Beta-Binomial](https://en.wikipedia.org/wiki/Beta-binomial_distribution)
+
+
+
+

docs/gallery_content.rst

@@ -408,7 +408,7 @@ Discrete Distributions
       Bernoulli
 
    .. grid-item-card::
-      :link: ./api_reference.html#preliz.distributions.betabinomial.BetaBinomial
+      :link: ./examples/gallery/betabinomial.html
       :text-align: center
       :shadow: none
       :class-card: example-gallery