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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.
 Summer '08
 MATTHEWTILL

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