In a caf, coffee and tea are on average sold at 3 cups and 1 cup respectively every 5 minutes. Let C and 7 denote the number of cups of coffee and tea sold in an hour interval. (a) State any assumptions required for us to use the Poisson distribution to model the random variables C and T. Comment on how reasonable the assumption is in practice. (b) (4 marks) (c) It is known that from 15:00 to 16:00, 45 cups of beverages are sold. Find the probability that the number of cups of coffee sold in this time period is between 30 to 33 cups, inclusive. State any assumptions you have made in your computations. (d) (4 marks) Assume that we can model C and T after Poisson distributions. Find the probability that a total of 45 cups of beverages (you may assume the only beverages sold in the caf are either coffee or tea) in a randomly chosen one hour interval. State any assumptions you have made in your computations. (e) (6 marks) Let W denotes the time (in hours) passed after a cup of beverage is sold before the next is sold. State the probability distribution of W and write down the values of E(W) and Var(W) (5 marks) I am in a queue at the caf waiting to order my drink. There are two counters, A and B, in front of me that I can turn to. On average, I need to wait for 10 minutes for my turn to arrive to order at counter 4 and 8 minutes for that at counter B. I will proceed to whichever counter is available first. Find the probability that I will have to wait for at least 7 minutes for my turn to make my order. State any assumptions you have made in your computations.