Consider a betting system with two possible outcomes labelled 1 and 2, and two possible wagers, also labelled 1 and 2. The return function is given by r₁ (1) = 2.4, r₁ (2) = 3.3, r₁ (2) = x, and r₂(2) = 0.7, where x is a real number. You are told that the betting system is fair, that is, there is no arbitrage. Determine x correct to 3 significant digits. Answer:
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- Three balanced coins are tossed independently. One of the variables of interest is Y₁, the number of heads. Let Y₂ denote the amount of money won on a side bet in the following manner. If the first head occurs on the first toss, you win $1. If the first head occurs on toss 2 or on toss 3 you win $2 or $3, respectively. If no heads appear, you lose $1 (that is, win -$1). a Find the joint probability function for Y₁ and Y2. b What is the probability that fewer than three heads will occur and you will win $1 or less? [That is, find F(2, 1).]The following dice game is offered to you. You may simultaneously roll one red die and three blue dice. The stake is $1. If none of the blue dice matches the red die, you lose your stake; otherwise, you get anyway paid k +1 dollars if exactlyk of the blue dice match the red die. In the case that exactly one blue die matches the red die, you get paid an additional $0.50 if the other two blue dice match. the median payoff of the game isConsider two possible wagers labelled 1 and 2, and two possible outcomes, also labelled 1 and 2. Three values of the return function are given by r₁ (1) = −3, r₁ (2) = 5, r₂ (1) = 4. Assume there are no arbitrage opportunities. Only one of the following statements about ₂ (2) is true. Which one is it? Select one: O a. r₂ (2) is a positive integer O b. O c. O d. O e. O f. O g. r₂ (2) cannot be determined by the information given 1₂ (2) is a negative non-integer fraction r₂ (2) = 0 none of the other statements are true r₂ (2) is a negative integer 1₂ (2) is a positive non-integer fraction
- Suppose you are deciding whether or not to bring an umbrella to school and the weather can be either sunny or rainy. Initially, you think the probability that it will be sunny is P(S)=0.6, and the probability that it will be rainy is P(R)=0.4. If you bring an umbrella and it rains, your payoff is 6. If you bring an umbrella and it is sunny, your payoff is 9. If you don’t bring an umbrella and it rains, your payoff is 0. If you don’t bring an umbrella and it is sunny, your payoff is 10. Suppose that you check two websites and see that both say it will be sunny. You know that the two forecasts are independent of each other, and each forecast is correct 70% of the time. That is, the probability that the website says it will be sunny given that it actually will be sunny is 0.7, and the probability that the website says it will rain given that it actually will rain is 0.7. If you are a Bayesian updater, what is your updated belief about the probability that it will be sunny? Do you bring…Avicenna, an insurance company, offers five-year commercial property insurance policies to small businesses. If the holder of one of these policies experiences property damage in the next five years, the company must pay out $23,600 to the policy holder. Executives at Avicenna are considering offering these policies for $791 each. Suppose that for each holder of a policy there is a 3% chance they will experience property damage in the next five years and a 97% chance they will not. (If necessary, consult a list of formulas.) If the executives at Avicenna know that they will sell many of these policies, should they expect to make or lose money from offering them? How much? To answer, take into account the price of the policy and the expected value of the amount paid out to the holder. O Avicenna can expect to make money from offering these policies. In the long run, they should expect to make dollars on each policy sold. O Avicenna can expect to lose money from offering these policies.…Avicenna, an insurance company, offers five-year commercial property insurance policies to small businesses. If the holder of one of these policies experiences property damage in the next five years, the company must pay out $26,500 to the policy holder. Executives at Avicenna are considering offering these policies for $497 each. Suppose that for each holder of a policy there is a 2% chance they will experience property damage in the next five years and a 98% chance they will not.(If necessary, consult a list of formulas.) If the executives at Avicenna know that they will sell many of these policies, should they expect to make or lose money from offering them? How much? To answer, take into account the price of the policy and the expected value of the amount paid out to the holder. Avicenna can expect to make money from offering these policies. In the long run, they should expect to makedollars on each policy sold. Avicenna can…
- A risk averse individual faces uncertainty with two outcomes: good, bad. Theindividual has income $1260 at good and $840 at bad outcome. The probability of good outcome is 5/7 (so the probability of bad outcome is 1 – 5/7 = 2/7). The individual can buy any nonnegative x units of insurance. Every unit of insurance has price $p and it pays $1 in the event of bad outcome. In this insurance market, the unit price of insurance is known to be p = 2/3. (a)Suppose the individual buys x units of insurance. Determine the individual's net income under good income, net income under bad income and the average net income. Draw these three in a diagram as functions of x. (b) For the individual: (i) compare full insurance with over insurance and (ii) compare full insurance with partial insurance. Then determine best choice of insurance for the individual.Suppose I flip a fair coin n = 20 times. A psychic tries to predict the outcome before each flip. Three researchers have different ideas about the psychic's ability. There is Sydney, the Skeptic (S), who thinks the psychic's success rate is between 49% and 51%. There is Morgan, the Mark, M, who thinks that the psychic's success rate is 80%. And there is Carter, the Cynic (C), who thinks the psychic's success rate is 10%. Specifically: S+ 0 ~ U(.49, .51) M + 0 = .80 C0 = .10 %3D In all cases, assume the number of successful predictions follows a binomial distribution with success rate 0. Usek for the number of successes and n for the number of trials. Given all that: Determine the formula for the Bayes factor (a.k.a., likelihood ratio) supporting Carter over Morgan. Call that Bayes factor Bc:M Determine the formula for the Bayes factor (a.k.a., likelihood ratio) supporting Morgan over Sydney. Call that Bayes factor BM:S Determine the formula for the Bayes factor (a.k.a., likelihood…Suppose you are deciding whether or not to bring an umbrella to school and the weathercan be either sunny or rainy. Initially, you think the probability that it will be sunny isP (S) = 0.6, and the probability that it will be rainy is P (R) = 0.4. If you bring an umbrellaand it rains, your payoff is 6. If you bring an umbrella and it is sunny, your payoff is 9. Ifyou don’t bring an umbrella and it rains, your payoff is 0. If you don’t bring an umbrellaand it is sunny, your payoff is 10. Suppose that you check two websites for weather forecasts.You know that the two forecasts are independent of each other, and each forecast is correct70% of the time. That is, the probability that the website says it will be sunny given that itactually will be sunny is 0.7, and the probability that the website says it will rain given thatit actually will rain is 0.7. Suppose that you suffer from confirmation bias. You initially thinkit is more likely to be sunny, so you misinterpret rain forecasts as sun…
- Rhett owns a cupcake bakery and is analyzing his sales of cupcake delivery orders. Based on his daily sales of delivery orders for the past month, he has already calculated the probabilities, ?(?),P(X), for the number of boxes of cupcakes, ?,X, purchased by a single customer in a single day, as shown in the table. ?X 1 2 3 4 5 ?(?)P(X) 0.75 0.05 0.10 0.05 0.05 Calculate the mean number of boxes of cupcakes, ??,μX, sold and delivered per person in a single day. Express your answer to two decimal places. ??=μX= If a box of cupcakes costs $40.00$40.00 and the flat-rate delivery fee is $7.00$7.00, calculate the mean sales per person, ??,μY, that the cupcake shop makes in a single day for delivery orders. Express your answer to the nearest cent. ??=$μY=$Suppose you are deciding whether or not to bring an umbrella to school and the weathercan be either sunny or rainy. Initially, you think the probability that it will be sunny isP(S)=0.6, and the probability that it will be rainy is P(R)=0.4. If you bring an umbrella and itrains, your payoff is 6. If you bring an umbrella and it is sunny, your payoff is 9. If you don'tbring an umbrella and it rains, your payoff is O. If you don't bring an umbrella and it is sunny,your payoff is 10. Now suppose that you suffer from confirmation bias. You are pretty sure that it is going tobe sunny, so you misinterpret rain forecasts as sun forecasts 20% of the time but alwayscorrectly interpret sun forecasts. You are aware of your own confirmation bias. If yourinterpretation is that you have seen two sun forecasts, what is your updated belief about theprobability of sun? Will you bring your umbrella?. The table below states the payoffs in political points (measured in billions of rubles) to two nations that are rivals in world politics, Russia and Ukraine. Each country can take one of two courses: peace; or war. In each cell, the first payoff is for Russia, and the second payoff is for Ukraine. (a) Assume that neither country observes the military strategy of its rival, and solve the game (if it can be solved). Explain your solution step-by-step. Does this outcome maximize total political points? (b) In general, what is a Nash equilibrium? Is the solution to this game a Nash equilibrium? (c) Suppose that each country deposits a fund of two billion rubles with the United Nations. Either country would forfeit this fund if it wages war. What is the solution now to the game? Is this a Nash equilibrium? Russia, Ukraine Peace War Peace 4,4 1, 5 War 5, 1 2,2
