Homework9 - HOMEWORK PROBLEMS#9 SOLUTIONS TO EVEN PROBLEMS...

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HOMEWORK PROBLEMS #9 SOLUTIONS TO EVEN PROBLEMS 38. Y U(0 , 1) (a) Then f ( y ) = ( 1 1 - 0 , 0 < y < 1 , 0 , elsewhere. and F ( y ) = Z y -∞ f ( t ) dt = 0 , y 0 y, 0 < y < 1 1 , y 1 . (b) Since 0 a + b 1 and 0 a 1, we have that P ( a Y a + b ) = F ( a + b ) - F ( a ) = ( a + b ) - a = b , which depends only on the value of b . 42. We want to find y such that F ( y ) = 0 . 5. Therefore, y - θ 1 θ 2 - θ 1 = 0 . 5, solving for y we get that the median is y = φ 0 . 5 = θ 2 + θ 1 2 . 44. (a) We must have that 1 = Z 2 - 2 kdy = ky | 2 - 2 = k (2 - ( - 2)) = 4 k . Therefore, k = 1 4 . (b) F ( y ) = Z y -∞ f ( t ) dt = 0 , y < - 2 R y - 2 1 4 dt = y +2 4 , - 2 y 2 1 , y > 2 . 52. Y U(50 , 70), so E ( Y ) = 50+70 2 = 60 and Var( Y ) = (70 - 50) 2 12 = 100 3 33 . 33. 68. (a) Y N(2 . 4 , . 8 2 ). Therefore, P ( Y > 3 . 0) = P ( Y - 2 . 4 . 8 > 3 . 0 - 2 . 4 . 8 ) = P ( Z > . 75) 0 . 2266. 70. (a) By independentce, P ( Y 1 > 3 . 0 , Y 2 > 3 . 0 , Y 3 > 3 . 0) = [ P ( Y > 3 . 0)] 3 [0 . 2266] 3 0 . 0116. (b) X Bin(3 , 0 . 2266), we want P ( X = 2) = 3 2 (0 . 2266) 2 (1 - 0 . 2266) 0 . 1191. 88. Y Exp(2 . 4) (a) P ( Y > 3) = S (3) = e - 3 / 2 . 4 0 . 2865. (b) P (2 < Y < 3) = F (3) - F (2) = (1 - e - 3 / 2 . 4 ) - (1 - e - 2 / 2 . 4 ) = e - 2 / 2 . 4 - e - 3 / 2 . 4 0 . 1481. 90. P ( Y > 5) = S (5) = e - 5 / 2 . 4 0 . 1245. Therefore, X Bin ( 10 , e - 5 / 2 . 4 ) and we want P ( X 1) = 1 - P ( X = 0) = 1 - 10 0 ( e - 5 / 2 . 4 ) 0 ( 1 - e - 5 / 2 . 4 ) 10 0 . 7355. 98. Y Exp(4). We want to find y such that P ( Y > y ) = 0 . 05. Therefore, e - y/ 4 = 0 . 05, so y = - 4 ln 0 . 05 11 . 98. 104. Y Exp(100), and P ( Y > 200) = S (200) = e - 200 / 100 0 . 1353, so
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