Suppose the agent is on a game show where she has to pick one of 3 boxes labelled A, B, C. The grand prize has been placed in one of the boxes randomly. If she selects the correct box, she wins. Otherwise she leaves empty handed. The agent's uncertainty is captured by a state space S = {A, B, C} where s ? S is the state of the world where box s contains the grand prize. Assume that the agent's prior is uniform (that is, it places probability 1 3 on each state of the world), and suppose the agent has picked box A. Suppose that the game show host tells the agent "Wait a minute, there's some news for you: the prize is not in box C. Would you like to stay with box A or pick a different box?". Most people would be indifferent between box A and box B at this point, thinking that the odds are 50-50 that the prize is in A or B. Much to the surprise of most people (including academics), that intuition is wrong mathematically:1 the Bayesian posterior probability that box B contains the prize is strictly higher than box A containing it. (It is entertaining to read up on people's reaction to this so-called "Monty Hall problem"). In this problem you are asked to determine the Bayesian posterior probabilities, using the tools we introduced in class. [Warning: feel free to read up on the Monty Hall problem and the suggested solutions online, but you are asked to solve the problem using only the tools introduced in class, and you will not find the required form of the answer in the majority of articles you see online]. (a) Model the game show host's news as a Blackwell experiment. Specifically, for each state s ? S, write the possible news as a distribution over messages (where the possible messages are "the prize is not in A", "the prize is not in B" and "the prize is not in C"). Your answer should constitute no more than a table of the form we used to represent Blackwell experiments in class (see the lecture notes). (b) Compute the Bayesian posterior probabilities over S

11.40 In the preliminary studies of a new drug, a pharmaceutical firm needs to obtain infor- mation on the relationship between the dose level and potency of the drug. In order to obtain this information. a total of 18 test tubes are inoculated with a virus culture and incubated for an appropriate period of time. Three test tubes are randomlyr assigned to each of six different dose levels. The 18 test tubes are then injected with the randome assigned dose level of the drug. The measured response is the protective strength of the drug against the virus culture. Due to a prob- lem with a few of the test tubes. only two responses were obtained for dose levels 4. 8. and 32. The data are given here: Dose level 2 4 8 16 32 64 Response 5,1 3 10. 14 15, 1'7 20, 21 , 19 23, 29 23, 31. 30 a. Plot the data. b. Fit a linear regression model to these data. c. From a plot of the residuals. does there appear to be a possible lack of t of the linear Iruadel'?I A biologist is studying a population of bacteria that initially started with 7000 cells and with initial instantaneous rate change -14 cells per second. It is predicted that the values of the second deriviative of the population function are between 0 and 0.028 cells per square second. Question 1 Use the above information and Taylor's Theorem, to find a linear model for the population function as a function of time 35 seconds. Linear Model = Question 2 According to the linearly model, 1. when will the population of bacteria disappear? if = :] second(s). Round your answer to 3 decimal places. 2. the accuracy of the prediction is approximately E number of cells. Round your answer to 3 decimal places. Question 3 If the biologist wants the linear model to accurately describe the population to within 0.25 cells, how long will the biologist be able to use the linear model to describe the population of the bacteria? Round your answer to 2 decimal places. Let A: Ill'OO MOII OMH a. Find the normalized eigenvectors and use them as columns in an orthogonal matrix C. b. Show that C'AC = D, where D is diagonal with the eigenvalues of A on the diagonal. c. Show that A = CDC'. 2. One of the core principles I want students to take away from the class is what we've called the "economic decision-making rule," and informs our theory of individual behavior. a. Write what the economic decision-making rule is and define marginal analysis. b. Define sunk costs and opportunity costs and discuss which of the two should be included when engaging in marginal analysis. c. Explain three contexts where we have seen the economic-decision make rule, interpret it in these contexts, and explain how they inform optimal decision-making (for example, a perfectly competitive firm sets P=MC to maximize profits, which is an application of the economic decision-making rule).2/14 question Jol 10. Step 1 of2 Correct Inco A statistics professor would like to build a model relating student scores on the first test to the scores on the second test. The test scores from a random sample of 21 students who have previously taken the course are given in the table. Test Scores Student First Test Grade Second Test Grade 2 73 64 70 4 50 5 73 70 6 68 65 70 68 72 67 65 10 75 63 65 13 73 14 45 91 92 65 78 Skip Try Similar