13. Let Ti, i = 1, 2, denote the first two interarrival times of a Poisson process X(s), s 0, with rate . (So,

13. Let Ti, i = 1, 2, denote the first two interarrival times of a Poisson process X(s), s ≥ 0, with rate α. (So, according to Proposition 3.5-1, both T1 and T2 have an exponential distribution with PDF f (t) = αe−αt, t > 0.)

Show that T1 and T2 are independent. (Hint. Argue that P(T2 > t|T1 = s) = P(No events in (s, s + t]|T1 = s), and use the third postulate in definition 3.4-1 of a Poisson process to justify that it equals P(No events in (s, s + t]).

Express this as P(X(s+t)−X(s) = 0) and use part (2) of Proposition 3.4-2 to obtain that it equals e−αt. This shows that P(T2 > t|T1 = s), and hence the conditional density of T2 given T1 = s does not depend on s.)