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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Question
Please assist with the following questions with the answers.
Transcribed Image Text: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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2Fe7e322dd-b0e4-46c1-b73a-249289fe2dc4%2Fuc97uph_processed.png&w=3840&q=75)
Transcribed Image Text: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)
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