From 14aec7f373152d975e8fea0df3c38855bef72694 Mon Sep 17 00:00:00 2001
From: aleicazatti <ale.icazatti@gmail.com>
Date: Wed, 3 Jul 2024 20:00:55 -0300
Subject: [PATCH] Add Exponential distribution page

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Exponential Distribution
+
+The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process, in which events occur continuously and independently at a constant average rate. It has a memorylessness property, which means that the probability of the event occurring in the next instant of time does not depend on how much time has already elapsed. It is characterized by the rate parameter $\lambda$ which is the reciprocal of the mean of the distribution.
+
+Many real-world phenomena can be modeled using the exponential distribution across various fields. For example, it is used in physics to model the time between decays of a radioactive atom, in medicine to model the time between seizures in patients with epilepsy, in engineering to model the time between failures of a machine, and in finance to model the time between changes in the price of a stock.
+
+In Bayesian modeling, the exponential distribution is often used as a prior distribution for scale parameters, which must always be positive.
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Exponential Distribution PDF
+---
+
+import matplotlib.pyplot as plt
+import arviz as az
+from preliz import Exponential
+az.style.use('arviz-doc')
+lambdas = [0.5, 1., 2.]
+
+for lam in lambdas:
+    Exponential(lam).plot_pdf(support=(0, 5))
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Exponential Distribution CDF
+---
+
+for lam in lambdas:
+    Exponential(lam).plot_cdf(support=(0, 5))
+```
+
+## Key properties and parameters:
+
+```{eval-rst}
+========  ==========================================
+Support   :math:`x \in (0, \infty)`
+Mean      :math:`\frac{1}{\lambda}`
+Variance  :math:`\frac{1}{\lambda^2}`
+========  ==========================================
+```
+
+**Probability Density Function (PDF):**
+
+$$
+f(x|\lambda) = \lambda e^{-\lambda x}
+$$
+
+**Cumulative Distribution Function (CDF):**
+
+$$
+F(x|\lambda) = 1 - e^{-\lambda x}
+$$
+
+```{seealso}
+:class: seealso
+
+**Common Alternatives:**
+
+- [Gamma Distribution](gamma_distribution.md) - The Exponential distribution is a special case of the Gamma distribution with the shape parameter $\alpha = 1$.
+- [Weibull Distribution](weibull_distribution.md) - The Exponential distribution is a special case of the Weibull distribution with the shape parameter $\alpha = 1$.
+
+**Related Distributions:**
+
+- [Poisson Distribution](poisson_distribution.md) - While the Exponential distribution models the time between events in a Poisson process, the Poisson distribution models the number of events in fixed intervals of time or space.
+- [Geometric Distribution](geometric_distribution.md) - The discrete counterpart of the Exponential distribution, modeling the number of trials needed to achieve the first success in a sequence of Bernoulli trials.
+```
+
+## References
+
+- [Wikipedia - Exponential Distribution](https://en.wikipedia.org/wiki/Exponential_distribution)
+
+
+
+
+