ACTSC-232-1081-Midterm1_solutions

# ACTSC-232-1081-Midterm1_solutions - ± λ λ + x ² α ....

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ACTSC 232 – WINTER 2008 MID-TERM EXAM #1 - FEBRUARY 4 2008 NAME: I.D.: Aids: Calculator Time: 50 minutes Examiner: Johnny S.H. Li Question Maximum Mark 1 12 2 17 3 21 Total 50 1

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1. Given 2 p x = 0 . 98, p x +2 = 0 . 985 and 5 q x = 0 . 0775, calculate: (a) 3 p x . [4] 3 p x = 2 p x p x +2 = 0 . 98 × 0 . 985 = 0 . 96530 (b) 2 p x +3 . [4] 3 p x 2 p x +3 = 5 p x = 1 - 5 q x 2 p x +3 = 1 - 5 q x 3 p x = 1 - 0 . 0775 0 . 9653 = 0 . 95566 (c) 2 | 3 q x . [4] 2 | 3 q x = 2 p x - 5 p x = 0 . 98 - (1 - 0 . 0775) = 0 . 0575 2
2. The survival function for the complete future life time of a life age 0 is S T 0 ( t ) = 1 - t ω , t < ω. You are also given that 0 e 0 = 25. (a) Show that ω = 50 [6] 0 e 0 = Z ω 0 t p 0 dt = Z ω 0 S T 0 ( t ) dt = Z ω 0 1 - t ω dt = " t - t 2 2 ω # ω 0 = ω 2 = 25 . This implies ω = 50. (b) Calculate 20 p 40 . [4] 10 p 20 = S T 0 (30) S T 0 (20) = 1 - 30 50 1 - 20 50 = 2 3 . 3

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(c) Calculate Var[ T 10 ] [7] E( T 10 ) = Z ω - 10 0 t p 10 dt = Z 40 0 1 - t 40 dt = " t - t 2 80 # 40 0 = 20 . E( T 2 10 ) = Z ω - 10 0 2 t t p 10 dt = Z 40 0 2 t ± 1 - t 40 ² dt = Z 40 0 2 t - t 2 20 dt = " t 2 - t 3 60 # 40 0 = 533 . 33 . Var( T 10 ) = E( T 2 10 ) - [E( T 10 )] 2 = 133 . 33 . 4
3. The probability density function for the complete future lifetime of a life age 0 is f T 0 ( x ) = αλ α ( λ + x ) α +1 , α,λ > 0 . (a) Show that the survival function for a life age 0, S T 0 ( x ) is S T 0 ( x ) =

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Unformatted text preview: ± λ λ + x ² α . [6] S T ( x ) = 1-F T ( x ) = 1-Z x f T ( s ) ds = 1-Z x αλ α ( λ + s ) α +1 ds = λ α ( λ + x ) α 5 (b) Derive an expression for μ x . [5] μ x = f T ( x ) S T ( x ) = α λ + x Note that f T ( x ) and S T ( x ) are given in the question. 6 (c) Derive an expression for S T x ( t ). [5] S T x ( t ) = S T ( x + t ) S T ( x ) = ± λ λ + x + t ² α ± λ λ + x ² α = ³ λ + x λ + x + t ´ α . 7 (d) Using (b) and (c), or otherwise, ﬁnd an expression for f T x ( t ). [5] f T x ( t ) = S T x ( t ) μ x + t = ± λ + x λ + x + t ² α α λ + x + t . 8...
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## This note was uploaded on 03/20/2011 for the course ACTSC 232 taught by Professor Matthewtill during the Summer '08 term at Waterloo.

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ACTSC-232-1081-Midterm1_solutions - ± λ λ + x ² α ....

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