Question 2 (2.1) Prove the following: If Poisson process x_(t) with rate \lambda counts the arrivals of type 1 customers into a shop, and if Poisson process Y_(t) with rate \mu 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_(t) counting the total number of arrivals is also a Poisson process. What is the rate of the process Z_(t) ? (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.