11. Consider a Yule process starting with a single individual—that is, supposeX(0) = 1.

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

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

(b) Let X1, . . . ,Xj denote independent exponential random variables each having rate λ, and interpret Xi as the lifetime of component i. Argue that max(X1, . . . ,Xj) can be expressed as max(X1, . . . ,Xj) = ε1 + ε2 + · · · + εj where ε1, ε2, . . . , εj are independent exponentials with respective rates jλ,

(j − 1)λ, . . . , λ.

Hint: Interpret εi as the time between the i − 1 and the ith failure.

(c) Using

(a) and

(b) argue that P{T1 + · · · + Tj  t} = (1 − e−λt

)

j

(d) Use

(c) to obtain P1j(t) = (1 − e−λt

)

j−1 − (1 − e−λt

)

j = e−λt

(1 − e−λt

)

j−1 and hence, given X(0) = 1, X(t) has a geometric distribution with parameter p = e−λt .

(e) Now conclude that Pij(t) =



j − 1 i − 1



e−λti

(1 − e−λt

)

j−i