2. (25 points) For each statement below, circle the correct answer. You do NOT have to explain your answer. (a) (3 points) The rates coming out of a state in a transition rate diagram must sum to 1. i. This statement is True ii. This statement is False (b) (3 points) Let {N(t],t 2 0} be a Poisson process. Then for a given x > 0, P(N(4x) -N(2x) = mg = n) must equal P(N(3x) - N(x) = n). i. This statement is True ii. This statement is False (c) (3 points) All Continuous Time Markov Chains are birth and death processes. i. This statement is True 'I'I ii. This statement is a se (d) (4 points) The lifetimes of 3 cars C1, C2, and C3 are exponentially distributed with mean 10 years, 15 years, and 20 years, respectively. What is the probability that car C1 will break first? j, 0.1 ii. 0.222 iii. 0.333 iv. 0.462 v. 0.5 vi. None of the above (e) (4 points) IOE advisors have promised to respond to emails within 24 hours. Suppose the time it takes the advisors to respond to emails is exponentially distributed with mean 12 hours. If the advisors receive 100 emails during the first week of registration, how many emails do they expect to not answer within the promised 24 hours? i.0 ii. 13.54 iii. 14.28 iv. 50.00 v. None of the above (f) {4 points) Patients arrive at a clinic every 10 minutes on a Monday. Given that, on average, patients spend 30 minutes at the clinic, how many patients would you expect to see at the clinic at any given time on Monday? i. 0.33 patients ii. 3 patients iii. 180 patients iv. 300 patients v. None of the above (g) (4 points) Suppose that cookies are restocked at the school cafeteria according to a Poisson process at a rate of 4 restocks per hour. Let {N (t), t 2 0} be the number of restocking; that have happened up to time t. What is the probability that at least 2 restocking; happen during the first hour the cafeteria is open? 1. 0.3075 ii. 0.9034 iii. 0.9267 iv. 0.0916 v. None of the above