16. Independent random variables X and Y follow Poisson distributions with parameters ,tt1 and #2 respectively. Derive the distribution of the random variable X + Y. Lorry drivers call at random at a roadside transport cafe on a northsouth main road; the arrival rates are constant at 6 drivers per hour northbound and 2 drivers per hour southbound over the whole time that the cafe is open. Find the probabilities that, in a half- hour period, (a) 1 or more northbound and 1 or more southbound drivers call, (b) a total of at least 2 drivers call. The cafe manager has found that southbound drivers spend more than northbound ones, and that the cafe will run protably in a half-hour period provided that a total of at least 3 drivers or at least 2 southbound drivers call. What is the probability of making a prot in a half-hour period? {MPH