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Question

Let {X{t),t > 0} be a pure birth process such that λ_j = jλ, for j = 0 , 1 , . . . , where λ > 0. We suppose that X(0) = 1.

(a) Let T_n := min{t > 0: X{t) = n (> 1)}. That is, T_n is the time needed for the number of individuals in the population to be equal to n. Show that the probability density function of T_n is given by

(b) Let N(t) be the number of descendants of the ancestor of the population at time t, so that N(t) = X(t) - 1. Suppose that the random variable τ has an exponential distribution with parameter μ. Show that