81.

(a) Let {N(t),t 0} be a nonhomogeneous Poisson process with mean value function m(t). Given N(t) = n, show that the unordered set of arrival times has the same distribution as n independent and identically distributed random variables having distribution function F(x) =

⎧

⎨

⎩

m(x)

m(t) , x  t 1, x t

(b) Suppose that workmen incur accidents in accordance with a nonhomogeneous Poisson process with mean value function m(t). Suppose further that each injured man is out of work for a random amount of time having distribution F. Let X(t) be the number of workers who are out of work at time t. By using part (a), find E[X(t)].