7.10. Consider a stationary (0, A) Cox process.

(a) Show that Pr{N((0, h]) > 0, N((h, h + t]) = 0) = f(t; A) - f(t + h; A), whence

(c) We interpret the limit in

(b) as the conditional probability Pr (N((O, t]) = 01 Event occurs at time 0).
Show that Pr{N((0, t]) = 01 Event at time 01 = p+e-K+' + p_e-,U-', where

(d) Let The the time to the first event in (0,

c) in a stationary (0, A) Cox process with a = /3 = 1 and A = 2. Show that E[rIEvent at time 0] = 1.
Why does this differ from the result in Problem 7.4?

Pr{N((h, ht]) 0|N((0, h]) > 0} = f(t; )f(th; 1 - f(h; ) (b) Establish the limit where lim Pr{N((h, h + t]) = 0|N((0, h]) > 0} = h-0 df(t; ) f'(t; ) = dt f'(t; ) f'(0; )'