---
title: "Poisson mechanics"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```








*  Let $U_1$ and $U_2$ be independent and uniformly distributed on $[0,1]$.   Consider their order statistics $U_{(1)} < U_{(2)}$; what is their joint pdf?



* Prove that for a Poisson point process on $[0, \infty)$,  constructed with exponential inter-arrival times, that  conditional that there exactly *two* points in $[0,1]$, then those points are independently and uniformly distributed.




* Demonstrate this fact by simulations. Think about what you need to illustrate with the simulation.  


### Endnotes

* Version: `r format(Sys.time(), '%d %B %Y')`
* [source](https://tsoo-math.github.io/ucl6/week9-2025-inclass.Rmd)


