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.