FinalCS440, Fall 2003This test is closed book, closed notes, no calculators. You have 3:00 hours to answer the questions. If you think aproblem is ambiguously stated, state your assumptions and solve the problem under those assumptions. You can useboth sides of the test book to write your answers.Name:ID:Problem Score Max. score1 212 213 284 165 14Total 10011 Bayesian networksConsider the Bayesian network shown in Figure 1. Assume all random variables are binary. Nodes,, and haveABCDEFigure 1: Bayesian network for Problem 1.the same prior distribution,        . Nodehas a conditional Bernoulli(binomial) distribution   with parameters   . Nodealso has a conditional Bernoulli distribution but with parameters   .Note: In the following problems you do not need to compute1. [3 pts] Write the expression for the joint distribution defined by this network.2. [3 pts] List the nodes that belong to Markov blankets of nodes, and.3. [3 pts] What is ? Show your work.24. [4 pts] What is ? Show your work.5. [3 pts] What is ? Show your work.6. [5 pts] A child node,, is added to node. It has a conditional distribution defined by and   . Compute   . Show your work.32 Dynamic models and statistical learningJohn works in a wine cellar where he needs to implement a system for monitoring the levels of sugar in the wine. Hepurchased two sensors that return three discrete measurement corresponding to low, normal, and high levels of sugarand can be used to detect whether the grape mix is in normal or abnormal condition. However, the sensors are notperfect. The specification lists the following sensor characteristics:          1. [5 pts] John took a pair of measurements with the two sensors, at five different times. They were      and     !  . He knew nothing about what condition the mix was inbefore the measurements were taken. What is his best guess about the state of the mix during the measurementsif he assumes that all of the measurements were taken independently? Show the work that justifies your answer.2. [6 pts] John’s boss told him he should not really make a global decision like that. Rather, he should decide thecondition of the mix for each pair of measurements with the two sensors (i.e., John would have to make fivedecisions), after all the measurements were taken. But the boss also realized that the condition of the mix doesnot change abruptly after each pair of measurements is taken. Since he did not know any better he told John toassume the following sets of probabilities that relate the mix state at two consecutive times:normal atnormal at  normal atabnormal at What are the five decisions that John would make under these assumptions? Show your work.43. [5 pts] After seeing the results of John’s work, his boss told him he should come up with better estimates of thetransition probabilities. How could John do that?4. [5 pts] How would John use those new estimates to make better future decisions?53 Decision makingConsider three ways of computing the final score on an exam that consists ofquestions. One way is to averagescores of allquestions. Another one is to drop the lowest ofscores and then compute the average of scores ofthe otherquestions. Finally, one can assume that one of the questions will be counted towards extra credit andthe score will be computed by adding all the question scores and dividing the sum by.All problems are equally hard and it would take you an equal amount of time to solve each of them. The probabilityof correctly solving each of theproblems is.1. [4 pts] Assume the problems are independent and the probability of solving each individual problem does notdepend on how many other problems you can solve. What is the probability of correctly solvingout ofproblems? How many such eventsare there? Write the expression for this probability in terms of and.2. [4 pts] Assume the total score for the set ofproblems is". Each problem will be given the following score(reward, utility):Grading scheme 1 (GS1):"#if you solve it correctly and if you do not.Grading scheme 2 (GS2):"#if you solve it correctly, if you do not.Grading scheme 3 (GS3):"#if you solve it correctly, if you do not.What are the scores (utilities) of the three ways of grading? Explain your work.63. [6 pts] What are your expected scores under these three grading schemes? Show your work. You may want touse the fact that the expected value of the binomial distribution of trials with trial probabilityis .4. [3 pts] Which grading scheme one should choose? Justify your answer.75. [6 pts] What are the maximum differences between the expected scores of grading schemes GS2-GS1 andGS3-GS2 and when do they occur? Write your results in terms of .6. [5 pts] Analyze how the maximal differences depend on the student’s ability to solve the problems (probability) and the number of problems.84 Miscellaneous questions1. [5 pts] On your way out of the hit feature To Build a Decision Tree, you are surprised to find out the movietheater is giving away prizes. You watch the people ahead of you choose their prize either from behind Door #1or Door #2. Of those who chose Door #1, half received $6, 1% got a new bike worth $1000, and the rest got aworthless movie poster. Everyone who chose Door #2 got $13.Assuming you want to maximize the likely dollar value of your prize, what door should you choose? Why?2. Consider the joint probability distribution given by the table belowA B C P(A, B, C)False False False 0.05False False True 0.16False True False 0.03False True True 0.25True False False