Add Student's T distribution page

docs/examples/normal_distribution.md

@@ -78,7 +78,7 @@ F(x \mid \mu, \sigma) =
 \frac{1}{2} \left[ 1 + \text{erf} \left( \frac{x - \mu}{\sigma \sqrt{2}} \right) \right]
 $$
 
-where [erf](https://en.wikipedia.org/wiki/Error_function) is the error function.
+where erf is the [error function](https://en.wikipedia.org/wiki/Error_function).
 
 
 ```{seealso}

docs/examples/students_t_distribution.md

@@ -0,0 +1,102 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+
+# Student's t Distribution
+
+The Student's t distribution, also known as the t-distribution, is a continuous probability distribution that resembles the normal distribution but with heavier tails. It is characterized by its bell-shaped curve, symmetric around the mean, and can defined by three parameters: the degrees of freedom ($\nu$), the location parameter ($\mu$), and the scale parameter ($\sigma$). The smaller the value of ($\nu$), the heavier the tails of the distribution.
+
+It is often used in Bayesian analysis particulary as a robust alternative to the Normal due to the possibility of having heavier tails.
+
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Student's t Distribution PDF
+---
+
+import matplotlib.pyplot as plt
+import arviz as az
+from preliz import StudentT
+az.style.use('arviz-doc')
+nus = [2., 5., 5.]
+mus = [0., 0.,  -4.]
+sigmas = [1., 1., 2.]
+for nu, mu, sigma in zip(nus, mus, sigmas):
+    StudentT(nu, mu, sigma).plot_pdf(support=(-10,6))
+
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Student's t Distribution CDF
+---
+
+for nu in nus:
+    StudentT(nu, mu, sigma).plot_cdf(support=(-10,6))
+```
+
+## Key properties and parameters:
+
+```{eval-rst}
+========  ==========================================
+Support   :math:`x \in \mathbb{R}`
+Mean      :math:`\mu` for :math:`\nu > 1`, otherwise undefined
+Variance  :math:`\frac{\nu}{\nu-2}` for :math:`\nu > 2`,
+          :math:`\infty` for :math:`1 < \nu \le 2`, otherwise undefined
+========  ==========================================
+```
+
+**Probability Density Function (PDF):**
+
+$$
+f(x \mid \nu, \mu, \sigma) =  \frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}
+$$
+
+where $\Gamma$ is the [gamma function](https://en.wikipedia.org/wiki/Gamma_function).
+
+**Cumulative Distribution Function (CDF):**
+
+$$
+F(y \mid \nu, \mu, \sigma) = 
+\begin{cases} 
+1 - \frac{1}{2} I_{\frac{\nu}{x^2 + \nu}} \left( \frac{\nu}{2}, \frac{1}{2} \right) & \text{for } x = \frac{y - \mu}{\sigma} \leq 0, \\[0.5em]
+\frac{1}{2} I_{\frac{\nu}{x^2 + \nu}} \left( \frac{\nu}{2}, \frac{1}{2} \right) & \text{for } x = \frac{y - \mu}{\sigma} > 0,
+\end{cases}
+$$
+
+where $I_x(a, b)$ denotes the [regularized incomplete beta function](https://en.wikipedia.org/wiki/Regularized_incomplete_beta_function).
+
+
+
+```{seealso}
+:class: seealso
+
+**Common Alternatives:**
+
+- [Skewed Student's t Distribution](skew_studentt_distribution.md) - Extends the Student's t-distribution by introducing a skewness parameter, allowing for the modeling of data that is not symmetrically distributed. 
+- [Half-Student's t Distribution](halfstudentt_distribution.md) - Considers only the positive values of the Student's t-distribution. 
+- [Normal Distribution](normal_distribution.md) - When $\nu \to \infty$, the t-distribution converges to the normal distribution.
+- [Cauchy Distribution](cauchy_distribution.md) - The Cauchy distribution is a special case of the Student's t-distribution with $\nu=1$.
+```
+## References
+
+- Wikipedia. [Student's t-distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution)
+
+

docs/gallery_content.rst

@@ -295,7 +295,7 @@ Continuous Distributions
       Skew-Normal
 
    .. grid-item-card::
-      :link: ./examples/student_t_distribution.html
+      :link: ./examples/students_t_distribution.html
       :text-align: center
       :shadow: none
       :class-card: example-gallery