Consider a Yule process starting with a single individual-that is, suppose X(0) 1.

Let 7, denote the time it takes the process to go from a population of size i to one of size i + 1.

(a) Argue that T, =1,..., are independent exponentials with respective rates i.

(b) Let X., X, denote independent exponential random variables each having rate, and interpret X, as the lifetime of component i. Argue that max (X,, X,) can be expressed as max(x1,...,x) = + & + + Ej where &, 2,, E, are independent exponentials with respective rates /, (-1)...... Hint: Interprete, as the time between the 1 and the ith failure.

(c) Using

(a) and

(b) argue that PITT 1) = (1-e)

(d) Use

(c) to obtain that Py(t) = (1-e)-1-(1-e) = e^(1-e-My-1 and hence, given X(0) = 1, X(r) has a geometric distribution with parameter p =

e.

(e) Now conclude that Pu(t) = (-)-(1-