Problem 4 (Poisson distribution) Assume a retailer chain has 2 branches. Let X1 and X2 be the number of customers arriving 10:00-11:00am on Mondays to each of the branches. Assume Xi~ (a) (b) P01530110.) for 1' = 1, 2 and that the demand to one is independent of demand to the other. Derive the probability density function of the total number of customers of the 2 branches. What distribution does this random variable have? What is the mean and variance of this random variable? In this question, suppose that you do not know the distribution of demand to each store but you do know that the total demand, Y, (the sum of demand to both stores) is a Poisson random variable with rate A1 + A2. Suppose that when a customer has to choose which store to go to, the customer chooses with probability A1/(A1 + A2) to go to store 1 and with the remaining probability to store 2. This way, the demand to store 1 is the number of customers that chose store 1 over store 2 (among the total number of customers). - What is P [X 1 = 1'|Y = 11]? (Think about the marginal distribution of X1 given Y = n). - What is now the distribution of the demand to store 1'? That is, what is P[X1 = 1'] '2' Same question for store 2. Now suppose, more realistically, that demand is positively correlated and the correlation coefcient is 0.8. What is the mean and variance of the total demand X1 + X2? Give your answer as a function of A1 and A2