Each bacterium in a colony splits into two identical bacteria after an exponential time of param- eter , which then split in the same way but independently. Let Xt denote the size of the colony at time t, and suppose X0 = 1. Show that the probability generating function (t) = E(zXt ) satisfies (t) = zet + t 0 e(ts)(t s)2 ds , and deduce that, for q = 1 et, and n = 1, 2, , P(Xt = n) = qn1(1 q)