1. Consider a nonhomogeneous Poisson process whose intensity function λ(t) is bounded and continuous. Show that such a process is equivalent to a process of counted events from a (homogeneous) Poisson process having rate λ, where an event at time t is counted (independent of the past) with probability λ(t)/λ; and where λ is chosen so that λ(s)<> for all s.
2. Let T1, T2, ... denote the interarrival times of events of a nonhomogeneous Poisson process having intensity function λ(t).
(a) Are the Ti independent?
(b) Are the Ti identically distributed?
(c) Find the distribution of T1.