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Exercises_-_Chapter_9_-_Solutions

# Exercises_-_Chapter_9_-_Solutions - ACTSCI 331 Life...

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Unformatted text preview: ACTSCI 331 - Life Contingencies I Exercises (Chapter 9) 1. We have 20 p xy = Pr ( T ( xy ) > 20) = Pr (min ( T ( x ) ;T ( y )) > 20) = Pr ( T ( x ) > 20 ;T ( y ) > 20) = IND Pr ( T ( x ) > 20) Pr ( T ( y ) > 20) = 20 p x 20 p y = (0 : 9) (0 : 8) = 0 : 72 , and 20 p xy = 1 & 20 q xy = 1 & Pr (max ( T ( x ) ;T ( y )) ¡ 20) = 1 & Pr ( T ( x ) ¡ 20 ;T ( y ) ¡ 20) = IND 1 & Pr ( T ( x ) ¡ 20) Pr ( T ( y ) ¡ 20) = 1 & 20 q x 20 q y = 1 & (1 & : 9) (1 & : 8) = 0 : 98 . 2. We can express Pr (20 < T ( xy ) < 50) as Pr (20 < T ( xy ) < 50) = Pr ( T ( xy ) > 20) & Pr ( T ( xy ) > 50) = 20 p xy & 50 p xy , where n p xy = Pr ( T ( x ) > n;T ( y ) > n ) = Z 1 n Z 1 n & 1 4 & 2 e & & ( t + s ) + 3 4 ¡ 2 e & ¡ ( t + s ) ¡ dsdt = 1 4 e & 2 &n + 3 4 e & 2 ¡n = 1 4 e & : 02 n + 3 4 e & : 04 n . (1) It follows that Pr (20 < T ( xy ) < 50) = 1 4 e & : 02(20) + 3 4 e & : 04(20) & & 1 4 e & : 02(50) + 3 4 e & : 04(50) ¡ = 0 : 311 , 1 We also have Pr (20 < T ( xy ) < 50) = Pr ( T ( xy ) < 50) & Pr ( T ( xy ) < 20) = 50 q xy & 20 q xy , where n q xy = Pr ( T ( x ) < n;T ( y ) < n ) = Z n Z n & 1 4 & 2 e & & ( t + s ) + 3 4 ¡ 2 e & ¡ ( t + s ) ¡ dsdt = 1 4 ¢ 1 & e & &n £ 2 + 3 4 ¢ 1 & e & ¡n £ 2 = 1 4 ¢ 1 & e & : 01 n £ 2 + 3 4 ¢ 1 & e & : 02 n £ 2 . Thus, Pr (20 < T ( xy ) < 50) = & 1 4 ¤ 1 & e & : 01(50) ¥ 2 + 3 4 ¤ 1 & e & : 02(50) ¥ 2 ¡ & & 1 4 ¤ 1 & e & : 01(20) ¥ 2 + 3 4 ¤ 1 & e & : 02(20) ¥ 2 ¡ , = 0 : 249 . 3. Under the independent assumption, &nd (a) Pr (the &rst death amongst ( x ) and ( y ) occurs between times 10 and 30 ). Pr (10 < min ( T ( x ) ;T ( y )) < 30) = Pr ( T ( xy ) > 10) & Pr ( T ( xy ) > 30) = 10 p xy & 30 p xy = IND 10 p x 10 p y & 30 p x 30 p y = 0 : 9 (1 & : 15) & : 5 (1 & : 45) = 0 : 49 . (b) Pr (at most one death amongst ( x ) and ( y ) occurs between times 10 and 30). 1 & Pr ( 2 deaths between 10 and 30 ) = 1 & Pr (10 < T ( x ) < 30 ; 10 < T ( y ) < 30) = IND 1 & Pr (10 < T ( x ) < 30) Pr (10 < T ( y ) < 30) = 1 & ( 10 p x & 30 p x ) ( 10 p y & 30 p y ) = 1 & (0 : 9 & : 5) (1 & : 15 & (1 & : 45)) = 0 : 88 . (c) Pr (the second death amongst ( x ) and ( y ) occurs between times 10 and 30)....
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Exercises_-_Chapter_9_-_Solutions - ACTSCI 331 Life...

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