Events occur according to a Poisson process N(t), t 0, with rate . An event for

Events occur according to a Poisson process N(t), t ≥ 0, with rate λ. An event for which there are no other events within a time d of it is said to be isolated.

That is, an event occurring at timey

d, y + d).

(a) Find the probability that the first event of the Poisson process is isolated.

(b) Find the probability that the second event of the Poisson process is isolated.

Let Is (t) be the number of isolated events in the interval (s, s +t).

(c) If s >d, find E[Is (t)|N(s +t +d) = n].

(d) If s >d, find E[Is (t)].