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Exercises_-_Chapters_10-11

Exercises_-_Chapters_10-11 - ACTSCI 331 Life Contingencies...

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ACTSCI 331 - Life Contingencies I Exercises (Chapters 10-11) 1. Consider a two-decrement model ( (1) accidental death and (2) nonaccidental death) with ° ° (1) ( x ) = 0 : 01 , x ± 20 ; ° ° (2) ( x ) = 1 100 ° x , 20 ² x < 100 . Calculate (a) f T (50) ;J (15 ; 1) ; f T (50) ;J (15 ; 2) ; ° ( ° ) (65) ; (Answers: e ° 0 : 01(15) 50 ° 15 50 (0 : 01) , e ° 0 : 01(15) 50 ° 15 50 1 100 ° 65 and 0 : 01 + 1 100 ° 65 ) (b) 10 p ( ° ) 40 ; 10 q (2) 40 ; 10 q (1) 40 ; (Answers: e ° 0 : 01(10) 60 ° 10 60 , 1 60 1 ° e ° 0 : 01(10) 0 : 01 and 1 ³ e ° 0 : 01(10) 60 ° 10 60 ³ 1 60 1 ° e ° 0 : 01(10) 0 : 01 ) (c) 15 j 10 q (2) 40 and P ( J = 2) = 1 q (2) 40 ; (Answers: 1 60 e ° 0 : 01(15) ° e ° 0 : 01(25) 0 : 01 and 1 60 1 ° e ° 0 : 01(60) 0 : 01 ) 2. We consider a three-decrement model ( (1) death, (2) withdrawal and (3) disability) with x q (1) x q (2) x q (3) x 30 0.01 0.20 0.005 31 0.01 0.20 0.004 32 0.02 0.20 0.002 . Assume that, for a speci°c medical study, a group of 1000 individuals of 30 years old is selected. Find (a) the expected number of survivors at age 31 and 32; (Answers: 1000 ´ 0 : 785 and 1000 ´ 0 : 785 ´ 0 : 786 ) (b) the expected number of those that will withdraw or become disabled in the next three years; (Answer: 1000 ´ f 0 : 205 + 0 : 204 ´ 0 : 785 + 0 : 202 ´ 0 : 785 ´ 0 : 786 g ) (c) the expected number of those that will die in the third year; (Answer: 1000 ´ 0 : 02 ´ 0 : 785 ´ 0 : 786 ) (d) the expected number of those that will die or become disabled in the third year. (Answer: 1000 ´ 0 : 022 ´ 0 : 785 ´ 0 : 786 ) 1
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3. We consider a two-decrement model with (1) 1 q (2) 50 = 0 : 80 (3) E [ T (50)] = 34 ; (2) 1 R 0 t t p ( ° ) 50 = 100 (4) ° (1) (50 + t ) = 0 : 01 , t > 0 .
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