*23. For a nonhomogeneous Poisson process with intensity function (), t > 0, where Jo () dt = 00, let X1, X2,... denote the sequence

*23. For a nonhomogeneous Poisson process with intensity function ë(ß), t > 0, where Jo ë(ß) dt = 00, let X1, X2,... denote the sequence of times at which events occur.

(a) Show that Jo1 A(0 dt is exponential with rate 1.

(b) Show that \x\_^(t) dt, / > 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.