homework_3_2005

homework_3_2005 - 0.0 0.0 5 1 12 32 31 19 21 0.3 0.2 0.1...

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Homework Set #3 Problem 1 Read Application Example 8 and do Problem 8.1. Problem 2 The way MIT admits undergraduate students is exemplified in the following table. Each applicant is rated to a discrete “scholastic index” X (horizontal axis) and a discrete “personal rating index” Y (vertical axis). The top number in each cell ( in bold ) is the number of applicants is a given year with the associated combination. The bottom number in each cell ( in italic ) is the probability of being accepted. (Although this is indeed the way MIT handles applications, all numbers are fictitious). Scholastic Index, X Personal Rating, Y 90-100 80-90 70-80 60-70 50-60 50 10 20 40 52 32 10 6 1.0 0.9 0.7 0.5 0.4 0.3 9 60 110 150 192 47 17 0.9 0.7 0.5 0.4 0.3 0.2 8 86 215 305 351 87 62 0.7 0.5 0.4 0.3 0.2 0.1 7 39 173 250 192 102 53 0.5 0.4 0.3 0.2 0.1 0.0 6 17 54 118 152 97 68 0.4 0.3 0.2 0.1
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Unformatted text preview: 0.0 0.0 5 1 12 32 31 19 21 0.3 0.2 0.1 0.0 0.0 0.0 (a) Plot the marginal PMF of the two indices. (b) Plot the conditional PMFs of (X|Y = 8) and (X|Y = 6). (c) Plot the conditional PMF of (Y|X 50). (d) What is the probability that an applicant with Y = 7 is accepted. (e) Are X and Y independent? Why? Problem 3 In Bounty Town, U.S.A., total precipitation during the crop-growing season, Q, has a uniform distribution between 2 and 4 inches. The total crop value $ depends on Q in such a way that ($|Q = q) has uniform distribution (in millions of dollars) between (2q 1) and (2q + 1). Note that the possible values of (Q,$) are inside the parallelepiped shaded in the figure below: 2q + 1 $ 2q - 1 2 4 q (a) What is the joint PDF of Q and $? (b) What is the marginal PDF of $? (c) What value of $ is exceeded on average every 5 years? Read Application Examples 7, 9 and 10. ....
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