A1Sol - ACTSC 232 Spring 2010 Assignment 1 SOLUTIONS 1 The...

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Unformatted text preview: ACTSC 232 - Spring 2010 - Assignment 1 SOLUTIONS 1. The function 20900- 80 x- x 2 20900 has been proposed for the survival function S ( x ) for a mortality model. (a) What is the implied limiting age ω ? Since S ( ω ) = 20900- 80 ω- ω 2 20900 = 0, we need ω 2 + 80 ω- 20900 = 0 so ( ω- 110)( ω + 190) = 0 so ω = 110 (negative age rejected) (b) Verify that S ( x ) satisfies the 3 conditions for a survival function. 1. S (0) = 20900- 80 *- 2 20900 = 1 . 2. lim x →∞ S ( x ) = S ( ω ) = 20900- 80 ω- ω 2 20900 = 0 . 3. d dx S ( x ) =- 2 x +80 20900 ≤ 0 so it is a decreasing function for x ≤ 110 (c) What is the pdf of the future lifetime random variable for a life age 0? f ( x ) =- d dx S ( x ) = 2 x +80 20900 = x +40 10450 for x ≤ 110. (d) Calculate 20 p . 20 p = S (20) = 20900- 80(20)- (20) 2 20900 = 0 . 90431. (e) Calculate 10 | 5 q . 10 | 5 q = 10 p- 15 p = (20900- 80(10)- (10) 2 )- (20900- 80(15)- (15) 2 ) 20900 = 0 . 02512. (f) Calculate the mean future lifetime for a life age 0. ◦ e = integraltext ω- x p dx = integraltext 110 20900- 80 x- x 2 20900 dx = [ x- 80 x 2 41800- x 3 62700 ] vextendsingle vextendsingle vextendsingle 110 = 65 . 61404 1 (g) Calculate the median future lifetime for a life age 0. We want to find m such that P ( T ≤ m ) = P ( T ≥ m ) = 0 . 5. So set S ( m ) = 0 . 5 and solve for m . 20900- 80 m- m 2 20900 = 0 . 5 m 2 + 80 m- 10450 = 0 m = 69 . 77249, using the quadratic formula and rejecting the negative root....
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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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A1Sol - ACTSC 232 Spring 2010 Assignment 1 SOLUTIONS 1 The...

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