Suppose that events occur according to a Poisson process with rate A. Each time an event occurs we must decide whether or not to stop, with our objective being to stop at the last event to occur prior to some specified time T. That is, if an event occurs at time t, 0 t T and we decide to stop, then we lose if there are any events in the interval (t, T], and win otherwise If we do not stop when an event occurs, and no additional events occur by time T, then we also lose Consider the strategy that stops at the first event that occurs after some specified time s, 0 s T.

(a) If the preceding strategy is employed, what is the probability of winning?

(b) What value of s maximizes the probability of winning?

(c) Show that the probability of winning under the optimal strategy is 1/e.