23. Let X(t) be a Poisson process with rate . (a) Use words to justify that the events [X(t) = 1] [X(1) = 1]

23. Let X(t) be a Poisson process with rate α.

(a) Use words to justify that the events

[X(t) = 1] ∩ [X(1) = 1] and

[X(t) = 1] ∩ [X(1) − X(t) = 0]

are the same

(b) Use Proposition 3.4-2 to find the probability of the event in

(a) when α = 2 and t = 0.6.

(c) It is given that X(1) = 1, that is, only one event in the time interval [0, 1]. Let T denote the time the event occurred, and let t be between 0 and 1.

(i) Use words to justify that the events T ≤ t and X(t) = 1 are the same.

(ii) Show that the conditional distribution of T, given that X(1) = 1, is uniform in [0, 1] by showing that P(T ≤ t|X(1) = 1) = t.