Age(x) lx dx Age(x) x dx 0 103 0 15 64 4 1 103 0 16 60 4 2345 2 103 3 17 56 7 100 1 18 49 10 4 99 1 19 39 6 98 6 95 7 93 8 92 9 90 10 86 11 82 3212443 20 33 3 21 30 10 22 23 4 24 8 4 25 26 12 79 4 27 13 75 3 28 14 72 8 29 843030 855221 20 8 12 4 1 1 Table 2: Actuarial table. 1. (Central exposed to risk) Suppose that the fractional part S = T-K, with K = [T], of lifetime T having distribution F is assumed to be uniformly distributed on [0, 1). This assumption may be useful in the interpolation/graduation of default probability for continuous lifetime. (a) Let f be density of T. Define Ax+s = f(x + s)/(1 − F(x + s)). Show for sЄ [0, 1) that 9x λx+s = Vx Є N. 1 - sqx (b) Use the result in 1(c) to get an estimate for λx+ in terms of dx and lx. (c) Deduce from eqn. (1) the central exposed to risk Ę aged x. (d) Use the Table 2, see page 2, and construct your estimate 7x+1½ and Ex. (1)
Age(x) lx dx Age(x) x dx 0 103 0 15 64 4 1 103 0 16 60 4 2345 2 103 3 17 56 7 100 1 18 49 10 4 99 1 19 39 6 98 6 95 7 93 8 92 9 90 10 86 11 82 3212443 20 33 3 21 30 10 22 23 4 24 8 4 25 26 12 79 4 27 13 75 3 28 14 72 8 29 843030 855221 20 8 12 4 1 1 Table 2: Actuarial table. 1. (Central exposed to risk) Suppose that the fractional part S = T-K, with K = [T], of lifetime T having distribution F is assumed to be uniformly distributed on [0, 1). This assumption may be useful in the interpolation/graduation of default probability for continuous lifetime. (a) Let f be density of T. Define Ax+s = f(x + s)/(1 − F(x + s)). Show for sЄ [0, 1) that 9x λx+s = Vx Є N. 1 - sqx (b) Use the result in 1(c) to get an estimate for λx+ in terms of dx and lx. (c) Deduce from eqn. (1) the central exposed to risk Ę aged x. (d) Use the Table 2, see page 2, and construct your estimate 7x+1½ and Ex. (1)
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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