Gallery: Add Censored and Truncated

docs/examples/censored_distribution.md

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Censored Distribution
+
+This is not a distribution per se, but a modifier of univariate distributions.
+
+A censored distribution arises when the observed data is limited to a certain range, and values outside this range are not recorded. For instance, in a study aiming to measure the impact of a drug on mortality rates it may be known that an individual's age at death is at least 75 years (but may be more). Such a situation could occur if the individual withdrew from the study at age 75, or if the individual is currently alive at the age of 75. Censoring can also happen when a value falls outside the range of a measuring instrument. For example, if a bathroom scale only measures up to 140 kg, and a 160-kg person is weighed, the observer would only know that the individual's weight is at least 140 kg.
+
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Censored Distribution PDF
+---
+
+import arviz as az
+from preliz import Normal, Censored
+az.style.use('arviz-doc')
+Censored(Normal(0, 1), -1, 1).plot_pdf(support=(-4, 4))
+Normal(0, 1).plot_pdf(alpha=0.5)
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Censored Distribution CDF
+---
+
+Censored(Normal(0, 1), -1, 1).plot_cdf(support=(-4, 4))
+Normal(0, 1).plot_cdf(alpha=0.5)
+```
+
+
+## Key properties and parameters:
+
+
+**Probability Density Function (PDF):**
+
+Given a base distribution with cumulative distribution function (CDF) and probability density mass/function (PDF). The pdf of a Censored distribution is:
+
+$$
+\begin{cases}
+    0 & \text{for } x < \text{lower}, \\
+    \text{CDF}(lower) & \text{for } x = \text{lower}, \\
+    \text{PDF}(x) & \text{for } \text{lower} < x < \text{upper}, \\
+    1-\text{CDF}(upper) & \text {for } x = \text{upper}, \\
+    0 & \text{for } x > \text{upper},
+\end{cases}
+$$
+
+where `lower` and `upper` are the lower and upper bounds of the censored distribution, respectively.
+
+**Cumulative Distribution Function (CDF):**
+
+The given expression can be written mathematically as:
+
+
+$$
+\begin{cases}
+    0 & \text{for } x < \text{lower}, \\
+    \text{CDF}(x) & \text{for } \text{lower} < x < \text{upper}, \\
+    1 & \text{for } x > \text{upper},
+\end{cases}
+$$
+
+where `lower` and `upper` are the lower and upper bounds of the censored distribution, respectively.
+
+
+```{seealso}
+:class: seealso
+
+
+**Related Distributions:**
+
+- [Truncated](truncated_distribution.md) - In a truncated distribution, values outside the range are set to the nearest bound, while in a censored distribution, they are not recorded.
+
+```
+
+## References
+
+- Wikipedia - [Censored distribution](https://en.wikipedia.org/wiki/Censoring_(statistics))

docs/examples/truncated_distribution.md

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+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+kernelspec:
+  display_name: Python 3
+  language: python
+  name: python3
+---
+# Truncated Distribution
+
+This is not a distribution per se, but a modifier of univariate distributions. 
+
+Truncated distributions arise in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or below a given threshold or within a specified range. For example, if the dates of birth of children in a school are examined, these would typically be subject to truncation relative to those of all children in the area given that the school accepts only children in a given age range on a specific date. There would be no information about how many children in the locality had dates of birth before or after the school's cutoff dates if only a direct approach to the school were used to obtain information. 
+
+## Probability Density Function (PDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Truncated Distribution PDF
+---
+
+import arviz as az
+from preliz import Gamma, Truncated
+az.style.use('arviz-doc')
+Truncated(Gamma(mu=2, sigma=1), 1, 4.5).plot_pdf()
+Gamma(mu=2, sigma=1).plot_pdf()
+```
+
+## Cumulative Distribution Function (CDF):
+
+```{code-cell}
+---
+tags: [remove-input]
+mystnb:
+  image:
+    alt: Trucated Distribution CDF
+---
+
+Truncated(Gamma(mu=2, sigma=1), 1, 4.5).plot_cdf()
+Gamma(mu=2, sigma=1).plot_cdf()
+```
+
+
+## Key properties and parameters:
+
+
+**Probability Density Function (PDF):**
+
+Given a base distribution with cumulative distribution function (CDF) and probability density mass/function (PDF). The pdf of a Truncated distribution is:
+
+$$
+\begin{cases}
+    0 & \text{for } x < \text{lower}, \\
+    \frac{\text{PDF}(x, dist)}{\text{CDF}(upper, dist) - \text{CDF}(lower, dist)}
+    & \text{for } \text{lower} <= x <= \text{upper}, \\
+    0 & \text{for } x > \text{upper},
+\end{cases}
+$$
+
+where `lower` and `upper` are the lower and upper bounds of the truncated distribution, respectively.
+
+**Cumulative Distribution Function (CDF):**
+
+The given expression can be written mathematically as:
+
+
+$$
+\begin{cases} 
+0 & \text{if } x_i < \text{lower} \\
+1 & \text{if } x_i > \text{upper} \\
+\frac{\text{CDF}(x_i) - \text{CDF}(\text{lower})}{\text{CDF}(\text{upper}) - \text{CDF}(\text{lower})} & \text{if } \text{lower} \leq x_i \leq \text{upper}
+\end{cases}
+$$
+
+where `lower` and `upper` are the lower and upper bounds of the truncated distribution, respectively.
+
+
+```{seealso}
+:class: seealso
+
+
+**Related Distributions:**
+
+- [Censored](censored_distribution.md) - In a censored distribution, values outside the range are not recorded, while in a truncated distribution, they are set to the nearest bound.
+- [TruncatedNormal](truncated_normal_distribution.md) - A truncated normal distribution is a normal distribution that has been restricted to a specific range.
+
+```
+
+## References
+
+- Wikipedia - [Truncated distribution](https://en.wikipedia.org/wiki/Truncated_distribution)