```{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 $\lambda$ ```

```{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. ```

```{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? ```

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

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