Question 2 (2.1) Prove the following: If Poisson process X, with rate A counts the arrivals of type 1 customers μ into a shop, and if Poisson process Y, with rate counts the arrivals of type 2 customers into the shop, and if the arrival times of the two types of customers are independent of each other, then the process Z, counting the total number of arrivals is also a Poisson process. What is the rate of the process Z? (2.2) Assume that a customer relations officer receives nice e-mails at the rate of 2 per hour, and nasty e-mails at the rate of 1 per hour. The e-mails all arrive independent of each other. Calculate the probabilities of the following events: (a) He receives no e-mails at all during a working day of 8 hours. (b) He receives only nice e-mails during the next 4 hours. (c) The next 3 e-mails are all nasty.

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