For a nonhomogeneous Poisson process with intensity function (t), t 0, where 0 (t)

For a nonhomogeneous Poisson process with intensity function λ(t), t ≥ 0, where

 ∞

0 λ(t) dt =∞, let X1,X2, . . . denote the sequence of times at which events occur.

(a) Show that

 X1 0 λ(t) dt is exponential with rate 1.

(b) Show that

 Xi Xi−1

λ(t) dt, i ≥ 1, are independent exponentials with rate 1, where X0 = 0.

In words, independent of the past, the additional amount of hazard that must be experienced until an event occurs is exponential with rate 1.