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Unformatted text preview: Answers to Practice Questions 5 – ACTSC 331, FALL 2011 1. The probability is Pr { T 40 < T 50 , 20 < T 50 < 30 } = Z 30 20 Pr { T 40 < t } f T 50 ( t ) dt == 0 . 0227 . 2. The probability is Pr { 2 < T xy ≤ 3 } = 2 p xy- 3 p xy = 0 . 17933 . 3. (a) We have Pr { T 30:40 > 20 | T 30 ≤ 50 } = Pr { T 30:40 > 20 , T 30 ≤ 50 } Pr { T 30 ≤ 50 } = Pr { T 40 > 20 , 20 < T 30 ≤ 50 } Pr { T 30 ≤ 50 } = 0 . 49124 . (b) The probability is given by Pr { T 40 < T 30 } = R ∞ Pr { T 40 < t } f T 30 ( t ) dt = 0 . 28084 . (c) We have V ar [ T 30:40 ] = 2 R ∞ t t p 30:40 dt- ( R ∞ t p 30:40 dt ) 2 = 376 . 46 . (d) We have E [ K 30:40 ] = e 30 + e 40- e 30:40 = ∞ X k =1 k p 30 + ∞ X k =1 k p 40- ∞ X k =1 k p 30:40 = 69 X k =1 70- k 70 + ∞ X k =1 e- . 01 k- 69 X k =1 70- k 70 ! e- . 01 k = 69 X k =1 70- k 70 + ∞ X k =70 e- . 01 k + 1 70 ( Ia ) 69 i = 106 . 42 , where i = e . 01- 1....
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