T1-232-2010W

# T1-232-2010W - (2 k-1 k p x(c Prove that e x = e x n n p x...

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ACTSC 232, WINTER 2010 Tutorial 1 – 3:30-4:20, Monday, January 18, in HH 1101 1. Assume that μ x = 2 100 - x for 0 x < 100. (a) Calculate the survival function S 0 ( x ) , x 0 for a newborn life. (b) Calculate the probability that (30) will die between ages 40 and 60. (c) Calculate the survival function t p x , t 0 for ( x ). (d) Calculate the probability density function of T 30 . (e) Calculate the expected future lifetime of (30). (f) Calculate the expected curtate future lifetime of (30). (g) Calculate Pr { K 30 = 20 } . (h) Calculate Pr { K 30 < 30 } . (i) Calculate Pr { K 30 40 } . 2. You are given that if the force mortality for ( x ) increases to μ x + t + c, 0 t 1 from μ x + t , 0 t 1, the probability that ( x ) will die in one year will double to 0.04. Calculate c . 3. Let K * x = min { K x , n } , where K x is the curtate future lifetime for ( x ) and n is a positive integer. (a) Derive the expression of Pr { K * x > k } in terms of the survival function of T x for all k = 0 , 1 , 2 ,... . (b) Denote E [ K * x ] by e x : n . Show that e x : n = E [ K * x ] = n X k =1 k p x and E [( K * x ) 2 ] = n X k =1

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Unformatted text preview: (2 k-1) k p x . (c) Prove that e x = e x : n + n p x e x + n . (d) Let n = 20 and assume that the force of mortality μ x follows De Moivre’s law with ω = 100. Calculate E [ K * 40 ] and V ar [ K * 40 ] . 4. You are given the following information about e x : x e x 75 10 . 4 76 10 . 77 9 . 5 Calculate the probability that a life aged 75 will be still alive at age 77. 1 5. Assume that l x = 8(100-x ) 2 , ≤ x < 100 . (a) Calculate μ 40 . (b) Calculate t p 40 , ≤ t < 60. (c) Calculate the probability that (40) will die in year 10. (d) Calculate the variance of the further lifetime of (40). 6. You are given the following life table: x l x q x d x 80 1000 200 81 0.2125 82 160 83 470 84 100 85 0.4 92 (a) Calculate the probability that (80) will be still alive at age 85. (b) Calculate 3 d 81 . 2...
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T1-232-2010W - (2 k-1 k p x(c Prove that e x = e x n n p x...

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