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**Unformatted text preview: **Seventh Homework Set — Solutions Chapter 5 Problem 1 (a) We have 1 = integraltext 1- 1 c (1- x 2 ) dx = cx parenleftBig 1- x 2 3 parenrightBig | 1- 1 = 4 3 c , so that c = 3 4 . (b) We have integraltext x- 1 f ( y ) dy = 3 4 y parenleftBig 1- y 2 3 parenrightBig | x- 1 = 1 2 + 3 4 x parenleftBig 1- x 2 3 parenrightBig if- 1 ≤ x ≤ 1. Hence, F ( x ) = x <- 1 , 1 2 + 3 4 x parenleftBig 1- x 2 3 parenrightBig- 1 ≤ x ≤ 1 , 1 x > 1 . Problem 2 Determine C : integraltext ∞ xe- x 2 dx =- 2 xe- x 2 | ∞ + integraltext ∞ 2 e- x 2 dx = (- 2 x- 4) e- x 2 | ∞ = 4, so that C = 1 4 . Now, we have P { X ≥ 5 } = integraltext ∞ 5 1 4 xe- x 2 =- ( x 2 + 1) e- x 2 | ∞ 5 = 7 2 e- 5 2 Problem 4 (a) P { X > 20 } = integraltext ∞ 20 10 x 2 dx =- 10 x | ∞ 20 = 1 2 . (b) F ( x ) = braceleftBigg x < 10 1- 10 x x ≥ 10 (c) Let’s assume that lifetimes of the six devices are independent of each other. Let p = 1- F (15). Then the desired probability is 6 summationdisplay...

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