Events occur according to a Poisson process of rate . Each time an event occurs, we will get a prize and we must decide whether or not to stop, with our objective being to stop at the last event to occur prior to some specied time T with T > 1/. 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 rst event to occur after some xed time s, 0 s T.

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

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

(c) Show that the probability of winning when using this strategy with the value of s specied in part (b) is 1/e.

(d) What is the expected number of prizes you will get under the optimal strategy?

*I know how to do (a) (b) (c), but I do not know How do deal with (d).