Question:

Buses arrive at a bus station with i.i.d. interarrival times following an exponential distribution with intensity ?. Alice arrives at the bus station at a deterministic time t. a. (4pts) What is the expected waiting time for Alice until next bus comes? b. (6pts) Let ? be the time when the last bus arrived before time t. Show that t ? ? follows an exponential distribution with parameter ?. c. (6pts) Show that the expected interarrival time between the last bus which arrived before time t and the first bus which arrives after time t is 2 ? . Explain why it is different from the general expected interarrival time 1 ? .

Consider a Markov chain {Xn, n = 0, 1, . ..} on the state space S = {0, 1, 2}. Suppose that the Markov chain has the transition matrix 2 10 10 10 2 P = 3 10 2 4 10 10 1. Show that the Markov chain has a unique stationary mass. 2. Let h denote the stationary mass of the Markov chain. Find h(x) for all x E S. 3. Show that the Markov chain has the steady state mass. 4. Let h* denote the steady state mass of the Markov chain. Find h*(x) for all x E S.The fact that the F'F'F usually bows away from the origin implies that ii? getting more of one good means getting more of another good. as the production of any good increases. there is an increase in the opportunity cost of producing it. getting more of one good means getting less of another good- as the production of any good increases. there is a decrease in the opportunity cost of producing it. Different people view things and outcomes differently. This is referred to as... Select one: O scarcity macroeconomics subjective values microeconomics rationing free goods rational decision-making scarce goods economics normative economics positive economics O ceteris paribus