5.6.5 Alpha particles are emitted from a fixed mass of material according to a Poisson process of rate . Each particle exists for a random duration and is then annihilated. Suppose that the successive lifetimes Y1;Y2; : : : of distinct particles are independent random variables having the common distribution function G.y/ D PrfYk  yg. Let M.t/ be the number of particles existing at time t.

By considering the lifetimes as markings, identify the region in the lifetime, arrival-time space that corresponds to M.t/, and thereby deduce the probability distribution of M.t/.