Exercise 1 (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 2 (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 3 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 4 (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 5 (Poisson on a disc) Simulate a Poisson point process of intensity \(100\) on a disc.