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3 solution.pdf - HW 3 Solution[100 points total Haoyang Liu...

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HW 3 Solution [100 points total] Haoyang Liu before lunar new year [For graders: some will use decimals and other fractions. I try to give both when it is not clear, but also check for yourselves.] [For students: please email me if there is an error.] problem 1 (a) θ (b) θ (c) θ (d) θ - θ 2 (e) 3 θ + 4 (f) 9( θ - θ 2 ) [grading scheme: 2 points each,12 points total] problem 2 [putting numbers in a table is also fine] (a) marginal for x: Pr ( x = 1) = 0 . 2 , Pr ( x = 2) = 0 . 5 , Pr ( x = 3) = 0 . 3. marginal for y: Pr ( y = 2) = 0 . 1 , Pr ( y = 3) = 0 . 3 , Pr ( y = 4) = 0 . 3 , Pr ( y = 5) = 0 . 3. (b) Pr ( y = 2 | x = 1) = 0 . 5 , Pr ( y = 3 | x = 1) = 0 . 5 Pr ( y = 3 | x = 2) = 0 . 4 , Pr ( y = 4 | x = 2) = 0 . 4 , Pr ( y = 5 | x = 2) = 0 . 2 (c) E ( y ) = 3 . 8 = 19 / 5 (d) cov ( x, y ) = E ( xy ) - ExEy = 8 . 5 - 3 . 8 * 2 . 1 = 0 . 52 [grading scheme: 2 points each, 8 points total] 1
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2 Problem 3 [putting numbers in a table is also fine] (a) Pr ( x = 1) = 1 / 3 , Pr ( x = 2) = 1 / 3 , Pr ( x = 3) = 2 / 9 , Pr ( x = 4) = 1 / 9, Pr ( x 1) = 1 / 3 , Pr ( x 2) = 2 / 3 , Pr ( x 3) = 8 / 9 , Pr ( x 4) = 1. (b) Pr ( z = 1) = 2 * 3 * 9 - 3 * 3 81 = 45 / 81 = 5 / 9 Pr ( z = 2) = 2 * 3 * 6 - 3 * 3 81 = 27 / 81 = 1 / 3 Pr ( z = 3) = 2 * 2 * 3 - 2 * 2 81 = 8 / 81 Pr ( z = 4) = 2 * 1 * 1 - 1 * 1 81 = 1 / 81 (c) E ( x ) = E ( y ) = 19 / 9 E ( z ) = 127 / 81 (d) Pr ( x = 1 , z = 1) = 27 / 81 Pr ( x = 2 , z = 1) = 9 / 81 , Pr ( x = 2 , z = 2) = 18 / 81 Pr ( x = 3
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