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hw6sol - 36-225 Homework 6 SOLUTIONS Grading Key 1 5 points...

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36-225 - Homework 6 SOLUTIONS + Grading Key 1. 5 points: 3 for finding E ( X 2 ) , 2 for V(X) Let X U ( a, b ). We already know that E ( X ) = a + b 2 . Then E ( X 2 ) = Z b a x 2 1 b - a dx = 1 b - a x 3 3 b a = 1 b - a b 3 - a 3 3 = 1 b - a ( b - a )( b 2 + ab + a 2 ) 3 = a 2 + ab + b 2 3 = V ( X ) = E ( X 2 ) - E 2 ( X ) = a 2 + ab + b 2 3 - ( a + b ) 2 4 = b 2 - 2 ab + a 2 12 = ( b - a ) 2 12 2. 6 points: 2 for each part. (a) P (475 < Y < 500) = Z 500 475 1 500 dy = 1 500 · y 500 475 = 500 - 475 500 = 1 20 (b) P (0 < Y < 25) = Z 25 0 1 500 dy = 25 500 = 1 20 (c) P (0 < Y < 250) = Z 250 0 1 500 dy = 250 500 = 1 2 3. 4 points: 2 for E ( A ) , 2 for V ( A ) R U (0 , 1) = f R ( r ) = 1 0 < r < 1 Therefore, E ( A ) = E ( πR 2 ) = Z 1 0 πr 2 · 1 dr = π r 3 3 1 0 = π 3 ( 1 . 05). V ( A ) = V ( πR 2 ) = E ( πR 2 ) 2 - E 2 ( πR 2 ) = E ( π 2 R 4 ) - π 2 9 . Now, E ( π 2 R 4 ) = Z 1 0 π 2 r 4 · 1 dr = π 2 r 5 5 1 0 = π 2 5 . And so, V ( A ) = V ( πR 2 ) = π 2 5 - π 2 9 = 4 π 2 45 ( . 88). 1

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4. 5 points: 3 for standardizing, 2 for using the table Let Y be the amount of money spent on maintenance. Then Y N (400 , 20). P ( Y > 450) = P ( Y - μ σ > 450 - 400 20 ) = P ( Z > 2 . 5) = . 0062 5. 6 points: 3 for set-up, 3 for work Need to find c such that P ( Y > c ) = 0 . 1. 0 . 1 = P ( Y > c ) = P ( Y - μ σ > c - 400 20 ) = P ( Z > c - 400 20 ) The closest number in the body of the table to 0.1 is 0.1003 which is the table entry for z = 1 . 28. Then 1 . 28 = c - 400 20 = c = (1 . 28)(20) + 400 = 425 . 6 6. 11 points: 4 for *, 6 for **, 1 for putting everything together X N (0 , 1) f X ( x ) = 1 2 π e x 2 2 ( -∞ ≤ x ≤ ∞ ) E ( e 3 X + e - X 2 2 ) = * z }| { E ( e 3 X ) + ** z }| { E ( e - X 2 2 ) * E ( e 3 X ) = M X ( t = 3) since X N (0 , 1) z}|{ = e 3 . 0+3 2 · 1 2 = e 4 . 5 * * E ( e - X 2 / 2 ) = Z -∞ e x 2 / 2 · 1 2 π e - x 2 / 2 dx = Z -∞ 1 2 π e - x 2 dx = Z -∞ 1 2 π e - x 2 2 · 1 2 | {z } almost N ( μ =0 , σ 2 = 1 2 ) dx = 1 2 Z -∞ 1 2 π · 1 2 e - x 2 2 · 1 2 | {z } N ( μ =0 , σ 2 = 1 2 ) = 1 2 · 1 = 1 2
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hw6sol - 36-225 Homework 6 SOLUTIONS Grading Key 1 5 points...

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