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

```{exercise, name="Deterministic colouring"}
Let $\Pi$ a Poisson point process on $[0, \infty)$.  Suppose we colour the first arrival blue, and then next arrival red, and continue colouring the points in this alternating fashion.   Consider the point processes $\Gamma$  formed by considering only the blue points.  Is this a Poisson point process?  Explain.
```
<br>


```{exercise, name="Shop keeper"}
Suppose we model the number of customers that arrive at a high street shop on at particular day by a Poisson process of intensity $\lambda >0$, where $\lambda$ is measured in customers per hour.     We wish to estimate $\lambda$.    Suppose the shop is really high-end and on some days has no customers, on its $6$ hours of operations.  The shop keeper only keeps track of whether she had has any customers are not; that is, her records $x = (x_1, \ldots, x_n)$ are a binary sequence.  Find a consisent estimator for $\lambda$

```
<br>

```{exercise, Name="Uniforms"}
Suppose we $\Pi$ is a Poisson point process on $[0, \infty)$ of intensity $\lambda$.     Using the construction of $\Pi$ as exponential inter-arrival times, prove that conditioned on the event that the unit interval contains exactly one point, the distribution of the its location is uniform.    
```
 
<br>

```{exercise, name="Random deletion"}
Simulate a Poisson process of intensity $\lambda=2$, say with $10000$ arrivals.  Delete each arrival independently with probability $p=\tfrac{1}{2}$ to from a new *thinned* process.  Plot a histogram of  the inter-arrival times of the thinned process.  What should you see?  Why?
```

<br>

```{exercise, name="Poisson on a disc"}
Simulate a Poisson point process of intensity $100$ on a disc.
```

<br>
<br>

*  Version: `r format(Sys.time(), '%d %B %Y')`
* [Rmd Source](https://tsoo-math.github.io/ucl/QHW4.Rmd)

