5. Poisson processes (a) Give the probability that there are exactly 3 arrivals in a time interval of length 2 for a Poisson process of intensity A. (b) Give the expected inter-arrival time for a Poisson process of intensity 5. (c) Explain what it means that increments are stationary for a Poisson process.(2) Let N : [0, oo) - Z be (the counting function associated to) a Poisson process of intensity A and fix o E (0, co). Show that t - N(at) is a Poisson process and determine its intensity. Hint: Remember that a Poisson process is the unique process with its marginal distributions.Show that the superposition of m independent Poisson processes with rates 21 , ..., 2, results in a Poisson process with rate ) = 2 + ... + Am. Hint: Use the three defining properties of a Poisson counting process(i) (a) Define a Poisson process with rate 1. (b) Define a compound Poisson process. (ii) Identify the circumstances in which a compound Poisson process is also a Poisson process. (iii) The cumulative amount of claims reaching an insurance company is modelled using a compound Poisson process. (a) Explain why the compound Poisson process has the Markov property. (b) Comment on whether this seems reasonable for the given insurance model. (c) State whether the compound Poisson process is weakly stationary. (d) Explain whether you expect the cumulative insurance claims to follow a weakly stationary process