{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$formed by considering only the blue points. Is this a Poisson point process? Explain.  {exercise, name="Shop keeper"} Suppose we model the number of customers that arrive at a high street shop on at particular day by a Poisson process of intensity$\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. 

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