Distributions Gallery: Add Rice (#565)

* Distributions Gallery: Add Rice

* Update docs/examples/gallery/rice.md

---------

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

docs/examples/gallery/rice.md

@@ -0,0 +1,125 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Rice Distribution
+
+<audio controls> <source src="../../_static/rice.mp3" type="audio/mpeg"> This browser cannot play the pronunciation audio file for this distribution. </audio>
+
+The Rice distribution is the probability distribution of the magnitude of a circularly-symmetric bivariate normal random variable. It's characterized by two parameters: $v$, which represents the non-centrality parameter, and $\sigma$, the scale parameter. 
+
+The Rice distribution is often used in signal processing, particularly in the analysis of noisy signals, such as radar and communication systems.
+
+## Parametrization
+
+The Rice distribution has two alternative parameterizations: in terms of $v$ and $\sigma$, or in terms of $b$ and $\sigma$. The relationship between the two is given by:
+
+$$
+\begin{align*}
+b & = \frac{v}{\sigma}
+\end{align*}
+$$
+
+## Probability Density Function (PDF):
+
+::::::{tab-set}
+:class: full-width
+
+:::::{tab-item} Parameters $v$ and $\sigma$
+:sync: v-sigma
+```{jupyter-execute}
+:hide-code:
+
+from preliz import Rice, style
+style.use('preliz-doc')
+nus = [0., 0., 4.]
+sigmas = [1., 2., 2.]
+for nu, sigma in  zip(nus, sigmas):
+    Rice(nu, sigma).plot_pdf(support=(0,10))
+```
+:::::
+
+:::::{tab-item} Parameters $b$ and $\sigma$
+:sync: b-sigma
+
+```{jupyter-execute}
+:hide-code:
+
+bs = [0., 0., 2.]
+for b, sigma in zip(bs, sigmas):
+    Rice(b=b, sigma=sigma).plot_pdf(support=(0,10))
+```
+:::::
+::::::
+
+## Cumulative Distribution Function (CDF):
+
+::::::{tab-set}
+:class: full-width
+
+:::::{tab-item} Parameters $v$ and $\sigma$
+:sync: v-sigma
+```{jupyter-execute}
+:hide-code:
+
+for nu, sigma in  zip(nus, sigmas):
+    Rice(nu, sigma).plot_cdf(support=(0,10))
+```
+:::::
+
+:::::{tab-item} Parameters $b$ and $\sigma$
+:sync: b-sigma
+
+```{jupyter-execute}
+:hide-code:
+
+for b, sigma in zip(bs, sigmas):
+    Rice(b=b, sigma=sigma).plot_cdf(support=(0,10))
+```
+:::::
+::::::
+
+## Key properties and parameters:
+
+```{eval-rst}
+========  ==============================================================
+Support   :math:`x \in (0, \infty)`
+Mean      :math:`\sigma \sqrt{\pi /2} L_{1/2}(-\nu^2 / 2\sigma^2)`
+Variance  :math:`2\sigma^2 + \nu^2 - \frac{\pi \sigma^2}{2}`
+          :math:`L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)`
+========  ==============================================================
+```
+
+**Probability Density Function (PDF):**
+
+$$
+f(x|\nu, \sigma) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2 + \nu^2}{2\sigma^2}\right) I_0\left(\frac{x\nu}{\sigma^2}\right)
+$$
+
+where $I_0$ is the modified [Bessel function](https://en.wikipedia.org/wiki/Bessel_function) of the first kind.
+
+**Cumulative Distribution Function (CDF):**
+
+$$
+F(x|\nu, \sigma) = 1 - Q_1\left(\frac{x}{\sigma}, \frac{\nu}{\sigma}\right)
+$$
+
+where $Q_1$ is the [Marcum Q-function](https://en.wikipedia.org/wiki/Marcum_Q-function).
+
+```{seealso}
+:class: seealso
+
+**Related Distributions:**
+
+- [Normal](normal.md) - The Rice distribution is the magnitude of a bivariate normal distribution.
+```
+
+## References
+
+- Wikipedia - [Rice distribution](https://en.wikipedia.org/wiki/Rice_distribution)

docs/gallery_content.rst

@@ -271,7 +271,7 @@ Continuous Distributions
       Pareto
 
    .. grid-item-card::
-      :link: ./api_reference.html#preliz.distributions.rice.Rice
+      :link: ./examples/gallery/rice.html
       :text-align: center
       :shadow: none
       :class-card: example-gallery

preliz/distributions/rice.py

@@ -35,9 +35,9 @@ class Rice(Continuous):
 
     ========  ==============================================================
     Support   :math:`x \in (0, \infty)`
-    Mean      :math:`\sigma {\sqrt  {\pi /2}}\,\,L_{{1/2}}(-\nu ^{2}/2\sigma ^{2})`
-    Variance  :math:`2\sigma ^{2}+\nu ^{2}-{\frac  {\pi \sigma ^{2}}{2}}L_{{1/2}}^{2}
-                        \left({\frac  {-\nu ^{2}}{2\sigma ^{2}}}\right)`
+    Mean      :math:`\sigma \sqrt{\pi /2} L_{1/2}(-\nu^2 / 2\sigma^2)`
+    Variance  :math:`2\sigma^2 + \nu^2 - \frac{\pi \sigma^2}{2}`
+              :math:`L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)`
     ========  ==============================================================
 
     Rice distribution has 2 alternative parameterizations. In terms of nu and sigma