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

# Playing with renewal processes

In this exercise, we will demonstrate what happens when we look at a renewal process $N(t)$ for a large value of $t \approx 150$.  Consider the following simple *checks*:

* Estimate by simulations the distribution of $D_1=N(1) - N(0)$, $D_2=N(151) - N(150)$, and $D_3=N(161)  - N(160)$.  If $N$ is stationary, then $D_1, D_2, D_3$ should all have the same distribution
  * Run this check on a Poisson process, generated by exponential inter-arrival times
  * Run this check on a the renewal process with inter-arrival times where $X_i = 1+ U_i$, where $U_i$ are uniformly distributed on $[0,1]$.





#  Endnotes

* Version: `r format(Sys.time(), '%d %B %Y')`
* [Rmd Source](https://tsoo-math.github.io/ucl4/week12-2023-inclass.Rmd)
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