hw7_sol - (b) P -. 005 < Z < . 005 =...

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IEOR 172: Probability and Risk Analysis for Engineers, Fall 2007 Homework 7 Solution Chapter 5 Question 2 Z 0 f ( x ) dx = Z 0 cxe - x 2 dx = - 2 c Z 0 xe - x 2 = - 2 cxe - x 2 | 0 + Z 0 2 ce - x 2 dx = - 4 ce - x 2 | 0 = 4 c = 1 c = 1 4 P ( X > 5) = Z 5 1 4 xe - x 2 = - 1 2 xe - x 2 | 5 - e - x 2 | 5 = 7 2 e - 5 2 . Question 4 (a) Z 20 10 x 2 dx = - 10 x | 20 = 1 2 . (b) F ( y ) = Z y 10 10 x 2 dx = 1 - 10 y ,y 10; F ( y ) = 0 ,y < 10 . (c) Assuming independece of the events that the devices exceed 15 hours, since ¯ F (15) = 10 15 = 2 3 , then p = 6 X i =3 6 i ! ± 2 3 ² i ± 1 3 ² 6 - i . Question 10 (a) P { goes to A } = P(5 < X < 15 or 20 < X < 30 or 35 < X < 45 or 50 < X < 60) = 2 / 3 . (b) same as (a). Question 19 Let Z = X - 12 2 , then Z is a standard normal, and . 1 = P ± Z > c - 12 2 ² . From table 5.1, we have P( Z < 1 . 28) = . 9, so c - 12 2 = 1 . 28 c = 14 . 56 . 1
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Question 20 Let X denote the number in favor. Then, X is normalal with mean 65 and standard deviation 65 · . 35 = 4 . 77 (b) P { 59 . 5 S n 70 . 5 } = P ± 59 . 5 - μ σ S - μ σ 70 . 5 - μ σ ² = Φ ³ 70 . 5 - 65 4 . 77 ´ - Φ ³ 59 . 5 - 65 4 . 77 ´ = 2Φ(1 . 153) - 1 = . 751 . Question 22 (a) P( . 9 - . 005 < X < . 9 + . 005) = P ³ - . 005 . 003 < Z < . 005 . 003 ´ = P( - 1 . 67 < Z < 1 . 67) = 2Φ(1 . 67) - 1 = . 905 . Hence 9.5 percent will de defective. This is each will be defective with probability 1-.905=.095.
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Unformatted text preview: (b) P -. 005 &lt; Z &lt; . 005 = 2 . 005 -1 = . 99 . Since . 005 = . 995 , then . 005 = 2 . 575 = . 0019. Question 30 P { black | X = 5 } = P { X = 5 | black } P { black } P { X = 5 | black } P { black } + P { X = 5 | white } P { white } = 1 3 2 exp -(5-6) 2 2 9 1 3 2 exp -(5-6) 2 2 9 + 1 2 2 exp -(5-4) 2 2 4 (1- ) = 3 e-1 / 18 3 e-1 / 18 + 1- 2 e-1 / 8 = . 3153 . 3153 + . 4412(1- ) = 1 2 , therefore = . 5832. Question 32 (a) e-1 (b) e-. 5 2 Theoretical Exercises Question 12 (a) b + a 2 . (b) . (c) 1-e-m = 1 / 2 or m = (1 / ) log 2. Question 26 ( X-a ) / ( b-a ). 3...
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hw7_sol - (b) P -. 005 &amp;amp;lt; Z &amp;amp;lt; . 005 =...

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