Suppose that the number of events occurring in a given time period is a Poisson random variable with parameter λ. If each event is classified as a type i event with probability pi, i = 1, . . . , n, ∑ pi = 1, independently of other events, show that the numbers of type i events that occur, i = 1, . . . , n, are independent Poisson random variables with respective parameters λpi, i = 1, . . . , n.