C Suppose That Each Customer Will With Probability 0 85 Not Buy Rice Or With Probability 0 15 Buy A Bag Of Rice E 1 (90.02 KiB) Viewed 44 times
(c) Suppose that each customer will, with probability 0.85, not buy rice or, with probability 0.15, buy a bag of rice. Each customer acts independently. For simplicity, assume that the purchase of rice is done at the time the customer arrives at the supermarket. Suppose that there are only 3 bags of rice in the supermarket at the current moment. i. What is the expected time until all bags of rice are sold? (4 marks] ii. Suppose that rice will be restocked at time T, where T follows the exponential distribution with mean 2. What is the probability that the restock happens before the fourth customer who wants to buy rice arrives? (Hint: note that [****e xk-1e-Bx dx = (k – 1)! Bk for B > 0 and positive integer k.) [6 marks]
2 iii. Following the setting of part ii, let Z be the amount of time that there is no rice at the supermarket; Z = 0 if the restock happens before the last bag of rice is sold. Calculate = E(Z) and Var(Z). [8 marks] (d) Suppose that each customer will stay in the supermarket for exactly half an hour. What is the expected total staying time of customers by t= 2? (For example, suppose that two customers = have arrived by t = 2, one at t = 0.5 and one at t = 1.75. In this case, by t = 2, the first = = 2 customer has stayed for 0.5 hours and left, and the second customer has stayed for 0.25 hours 7 and is present. The total staying time is 0.75.) [12 marks] (Hint: first evaluate E(2:11) S:) for fixed t using double expectation, where N(t) is the number of customers arrived up to time t, and S; is the arrival time of the jth customer.) N(t i=
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