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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Poisson Distribution
+
+The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time (or space) if these events occur with a known constant mean rate and independently of the time since the last event.
+
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Poisson Distribution PDF
+---
+
+import arviz as az
+from preliz import Poisson
+az.style.use('arviz-doc')
+for mu in [0.5, 3, 8]:
+    Poisson(mu).plot_pdf();
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Poisson Distribution CDF
+---
+
+for mu in [0.5, 3, 8]:
+    Poisson(mu).plot_cdf();
+```
+
+## Key properties and parameters:
+
+```{eval-rst}
+========  ==========================
+Support   :math:`x \in \mathbb{N}_0`
+Mean      :math:`\mu`
+Variance  :math:`\mu`
+========  ==========================
+```
+
+**Probability Density Function (PDF):**
+
+$$
+f(x \mid \mu) = \frac{e^{-\mu}\mu^x}{x!}
+$$
+
+**Cumulative Distribution Function (CDF):**
+
+$$
+f(x \mid \mu) = \frac{\Gamma(x + 1, \mu)}{x!}
+$$
+
+where $\Gamma(x + 1, \mu)$ is the [upper incomplete gamma function](https://en.wikipedia.org/wiki/Incomplete_gamma_function).
+
+```{seealso}
+:class: seealso
+
+**Common Alternatives:**
+
+- [NegativeBinomial](negative_binomial_distribution.md) - The Negative Binomial is often used as an alternative the Poisson when the variance is greater than the mean (overdispersed data).
+- [ZeroInflatedPoisson](zero_inflated_poisson_distribution.md) - The Zero-Inflated Poisson is used when there is an excess of zero counts in the data.
+- [HurdlePoisson](hurdle_distribution.md) - The Hurdle Poisson is used when there is an excess of zero counts in the data.
+
+**Related Distributions:**
+
+- [Binomial](binomial_distribution.md) - The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed. See [law of rare events](https://en.wikipedia.org/wiki/Poisson_distribution#law_of_rare_events). 
+- [Normal](normal_distribution.md) - For sufficiently large values of $\mu$, the normal distribution with mean $\mu$ and standard deviation \sqrt{\mu} can be a good approximation to the Poisson. 
+```
+
+## References
+
+- [Wikipedia - Poisson Distribution](https://en.wikipedia.org/wiki/Poisson_distribution)
+
+
+
+
+