3. (10 marks) Suppose that a theater starts checking tickets 10 minutes before the movie starts....

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3. (10 marks) Suppose that a theater starts checking tickets 10 minutes before the movie starts. More- over, the audience enter the movie theater in accordance with a Poisson process with rate 30 people per minute. What is the expected total waiting time for all the audience till the movie starts? Hint: Given the number of arrivals at time t, i.e., N(t) = n, the n arrival times S,..., S, have the same distribution as the order statistics corresponding to the n i.i.d. random variables uniformly distribut- ed on (0, t). 3. (10 marks) Suppose that a theater starts checking tickets 10 minutes before the movie starts. More- over, the audience enter the movie theater in accordance with a Poisson process with rate 30 people per minute. What is the expected total waiting time for all the audience till the movie starts? Hint: Given the number of arrivals at time t, i.e., N(t) = n, the n arrival times S,..., S, have the same distribution as the order statistics corresponding to the n i.i.d. random variables uniformly distribut- ed on (0, t).