Suppose that each event of a Poisson process with rate A is classified as being either of type 1, 2, , k If the event occurs at s, then, indepen- dently of all else, it is classified as type i with probability P,(s), i = 1, .., k, P(s) = 1. Let N,(t) denote the number of type i arrivals in [0, 1] Show that the N,(t), i = 1,, k are independent and N,(t) is Poisson distributed with mean A

f, P.(s) ds