Events occur according to a Poisson process of rate .Every time an event occurs, we will get a prize and we must decide whether to

Events occur according to a Poisson process of rate λ. Every time an event occurs, we will get a prize and we must decide whether to stop or not, our goal being to stop at the last event that occurs before a specific time T with T > 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 at time T, and we lose otherwise. If we don't stop when an event occurs and no additional events occur at time T, then we lose. Furthermore, if no events occur at time T, then we lose. Consider the strategy that stops at the first event that occurs after a 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 the probability of winning using this strategy with the value of s specified in part (b) is 1/e.

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