Tutorial1 - The graph of a piecewise survival function S (...

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Tutorial 1—ACTSC 232, Fall, 2011 1. You are given the following probability distribution function (cdf) for T 0 , the future lifetime of a newborn (0): F 0 ( t ) = 0 , t < 0 0 . 005 t, 0 t < 60 0 . 3 + 0 . 0175( t - 60) , 60 t < 100 1 , t 100 . (a) Verify that S 0 ( t ) 1 - F 0 ( t ) is a proper survival function. Is S 0 ( t ) differentiable at every point? (b) Calculate the probability that the person will die within 40 years. (c) Calculate the probability that the person will die between ages 40 and 80. (d) Calculate the probability that the person will die between ages 40 and 80, given that the person survives to age of 40. (e) Use actuarial notations to denote the probabilities in (b), (c) and (d). 2. Given F 0 ( t ) = (1 - 1 1+ t ) 10 for t 0, find expressions for: (a) S 0 ( t ) (b) S x ( t ) and calculate: (c) p 20 (d) 10 | 5 q 30 . 3. (SOA Exam M Fall 2005)
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Unformatted text preview: The graph of a piecewise survival function S ( x ) consists of 3 line segments with end-points (0 , 1), (25 , . 50), (75 , . 40), (100 , 0). Calculate 20 | 55 q 15 55 q 35 . 4. Assume that the force of mortality for a survival model is given by x = 1 110-x . (a) Find the survival function S ( t ) corresponding to this force of mortality, simpli-fying as far as possible. (b) What is the limiting age for this model? (c) Sketch the survival function S ( t ). (d) Find the density function f ( t ). (e) What do you notice about this survival model? (In particular, assess its suitability as a survival model for human mortality). 1...
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