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6. Smith has a small booth where he sells lottery tickets. Customers arrive according to a Poisson process of rate

λ per minute. He will close the shop on the first occasion that a minutes have elapsed since the last customer arrived. Let X be the amount of time that he keeps the shop open. Show that EX = λ−1(eλa − 1) minutes.

Hint: condition on the arrival time of the first customer. 


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