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- Ana’s preferences are consistent with expected utility. Denote by u(x) her utility function for a monetary outcome £x, and suppose that she is strictly risk-averse. Ana currently faces the monetary lottery which pays: £A with probability pA; £B with probability pB; and £C with probability pC. Now suppose that Ana is made the following o§er. For each realization of the original lottery, another lottery will be executed according to which she wins an additional poundwith probability 1/2 and loses a pound with probability 1/2. Describe the lottery that Ana faces if she accepts the o§er by describing the new payo§s and probabilities over these payo§s. (b) Show that she rejects the o§er and explain why.A statistician has to decide on the basis of one obser-vation whether the parameter θ of a Bernoulli distribu-tion is 0, 1 2 , or 1; her loss in dollars (a penalty that isdeducted from her fee) is 100 times the absolute valueof her error.(a) Construct a table showing the nine possible values ofthe loss function. (b) List the nine possible decision functions and con-struct a table showing all the values of the corresponding risk function. (c) Show that five of the decision functions are not admis-sible and that, according to the minimax criterion, the remaining decision functions are all equally good.(d) Which decision function is best, according to the Bayes criterion, if the three possible values of the param-eter θ are regarded as equally likely?Consider an individual who is risk-loving instead of risk-averse. a. Is U(I) concave or convex?b. Suppose this person is offered an actuarially fair insurance product that guarantees her a certain income, E[I]. Graph the consumer surplus this person receives from buying this insurance as p, the probability of being sick, varies from 0 to 1. You should plot p on the horizontal axis and consumer surplus on the vertical axis.c. Suppose, finally, that this person is offered a subsidy (perhaps from her parents) for buying insurance so that, if she buys insurance, she will be guaranteed an income γ E[I], where γ >1. With the subsidy, insurance is now actuarially unfair in her favor. Graph how her consumer surplus (M) changes as p, the probability of being sick, varies from 0 to 1. [Hint: draw a coordinate plane with p on the x-axis and M on the y-axis.] Based on this graph, under what conditions is she least likely to buy the subsidized insurance?
- The cashier line of a canteen can facilitate up to 60 customers an hour. Frequenters of the canteen arrive at an average of 50 an hour. Suppose that management wants to evaluate the desirability of opening a second order-processing station so that two customers can be served simultaneously. Assume a single waiting line with the first customer in line moving to the first available server. a. What is the average arrival time in minutes of customers? b. What is the average service time in minutes of the canteen? c. What is the probability that the canteen has a customer?Figures obtained from a city's police department seem to indicate that of all motor vehicles reported as stolen, 63% were stolen by professionals (P), whereas 37% were stolen by amateurs (primarily for joy rides) (A). Of the vehicles presumed stolen by professionals, 25% were recovered within 48 hours (R<48), 23% were recovered after 48 hours (R>48), and 52% were never recovered (N). Of the vehicles presumed stolen by amateurs, 33% were recovered within 48 hours, 63% were recovered after 48 hours, and 4% were never recovered. (b) What is the probability that a vehicle stolen by a professional in this city will be recovered within 48 hours?(c) What is the probability that a vehicle stolen in this city will never be recovered? (Round your answer to four decimal places.)Take the Gamblers ruin on [0, N] with a fair coin, i.e., a simple random walk with absorbing boundaries at 0 and N. Let T be the time which elapses before the SRW is absorbed by one of the boundaries (i.e., the duration of the game. Show that P(T < ∞) = 1 and Epecation (T^k) < ∞ for all k.
- The cashier line of a canteen can facilitate up to 60 customers an hour. Frequenters of thecanteen arrive at an average of 50 an hour. Suppose that management wants toevaluate the desirability of opening a second order-processing station so that twocustomers can be served simultaneously. Assume a single waiting line with the firstcustomer in line moving to the first available server.a. What is the average arrival time in minutes of customers?b. What is the average service time in minutes of the canteen?c. What is the probability that the canteen has a customer?d. What is the probability that the canteen does not have any customer?e. How long would the line be on average (average number of customers in thesystem)?f. How many people are waiting to be served on average?g. How long in minutes would it take the customer from lining up until he leaves thewaiting line?h. How long in minutes would a customer wait to be served on average?i. Find the probability that there are 7 customers in the…In a recent study, 50males used a new weight-loss supplement, and 38 of them experienced weight loss after two weeks. In the same study, 20 females used the same supplement, and 16of them experienced weight loss after two weeks.Fill in t he blanks below to make the most reasonable statement possible. The new weight-loss supplement was less effective on ▼(Choose one: males/females) in the study. That is because only ____% of them lost weight after two weeks, whereas ____% of the ▼(Choose one:males/females) lost weight after two weeks.In a firm, the cost of producing two items of type A and one of type B is $200. If it is known that by producing one item of type A and three of type B the cost it's $150. Determine, using Cramer's rule, the unit cost of each article.
- A new warehouse is being designed and a decision concerning the number of loading docks is required. There are two models based on truckarrival assumptions for the use of this warehouse, given that loading a truck requires 1 hour. Using the first model, we assume that the warehouse could be serviced by one of the many thousands of independent truckers who arrive randomly to obtain a load for delivery. It is known that, on average, 1 of these trucks would arrive each hour. For the second model, assume that the company hires a fleet of 10 trucks that are assigned full time to shipments from this warehouse. Under that assumption the trucks would arrive randomly, but the probability of any truck arriving during a given hour is 0.1. Obtain the appropriate probability distribution for each of these assumptions and compare the results.A consumer is given the chance to buy a baseball card for $1, but hedeclines the trade. If the consumer is now given the baseball card, willhe be willing to sell it for $1? Standard consumer theory suggests yes, butbehavioral economists have found that “ownership” tends to increase thevalue of goods to consumers. That is, the consumer may hold out for someamount more than $1 (for example, $1.20) when selling the card, eventhough he was willing to pay only some amount less than $1 (for example,$0.88) when buying it. Behavioral economists call this phenomenon the“endowment effect.” John List investigated the endowment effect in a randomized experiment involving sports memorabilia traders at a sports-card show. Traders were randomly given one of two sports collectibles, say good A or good B, that had approximately equal market value.1 Those receiving good A were then given the option of trading good A for good B with the experimenter; those receiving good B were given the option of…Suppose you arrive into a building and are about to take an elevator to the your floor. Once you call the elevator, it will take between 3 and 45 seconds to arrive to you. We will assume that the elevator arrives uniformly between 3 and 45 seconds after you press the button. Find σ.