Events occur according to a Poisson process with rate . Each time an event occurs, we must

Events occur according to a Poisson process with rate λ. 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 , whereT >1/λ. That is, if an event occurs at time t, 0 ≤ t ≤ T , and we decide to stop, then we win if there are no additional events by time T , and we lose otherwise. If we do not stop when an event occurs and no additional events occur by time T , then we lose. Also, if no events occur by time T , then we lose. Consider the strategy that stops at the first event to occur after some fixed time s, 0 ≤ s ≤ T .

(a) Using this strategy, what is the probability of winning?

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

(c) Show that one’s probability of winning when using the preceding strategy with the value of s specified in part

(b) is 1/e.