add logistic distribution page

docs/examples/logistic_distribution.md

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Logistic Distribution
+
+The logistic distribution is a continuous probability distribution with a shape that resembles the normal distribution, but with heavier tails. It is defined by two parameters: the mean ($\mu$) and the scale parameter ($s$). The mean determines the center of the distribution, while the scale parameter controls the steepness of the curve.
+
+Its cumulative distribution function is the [logistic function](https://en.wikipedia.org/wiki/Logistic_function), which is characterized by an S-shaped curve (sigmoid curve). It is particularly useful in modeling growth processes, such as population growth, where the rate of growth decreases as the population reaches its carrying capacity. 
+
+In logistic regression (whether frequentist or Bayesian flavor), the logistic distribution is used to model the probability of a binary outcome based on one or more predictor variables. The logistic function maps any real value into the range [0, 1], making it suitable for binary classification tasks. 
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Logistic Distribution PDF
+---
+
+import matplotlib.pyplot as plt
+import arviz as az
+from preliz import Logistic
+az.style.use('arviz-doc')
+mus = [0., 0., -2.]
+ss = [1., 2., .4]
+for mu, s in zip(mus, ss):
+    Logistic(mu, s).plot_pdf(support=(-5,5))
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Logistic Distribution CDF
+---
+
+for mu, s in zip(mus, ss):
+    Logistic(mu, s).plot_cdf(support=(-5,5))
+```
+
+
+## Key properties and parameters:
+
+```{eval-rst}
+========  ==========================================
+Support   :math:`x \in \mathbb{R}`
+Mean      :math:`\mu`
+Variance  :math:`\frac{\pi^2}{3}s^2`
+========  ==========================================
+```
+
+**Probability Density Function (PDF):**
+
+
+$$
+f(x \mid \mu, s) = 
+\frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2}
+$$
+
+**Cumulative Distribution Function (CDF):**
+
+$$
+F(x \mid \mu, s) = \frac{1}{1 + e^{-(x - \mu) / s}}
+$$
+
+```{seealso}
+:class: seealso
+
+**Common Alternatives:**
+
+- [Normal Distribution](normal_distribution.md) - Often used as an alternative to the logistic distribution when the tails are not of primary concern.
+- [Cauchy Distribution](cauchy_distribution.md) - Has much heavier tails than the logistic distribution, making it a robust alternative when outliers are a concern.
+
+**Related Distributions:**
+
+- [Log-logistic Distribution](log_logistic_distribution.md) - The probability distribution of a random variable whose logarithm has a logistic distribution.
+```
+
+## References
+
+- [Wikipedia - Logistic Distribution](https://en.wikipedia.org/wiki/Logistic_distribution)