Suppose that the fractional part S = T - K, with curtate lifetime K = [7], of lifetime T is assumed to be uniformly distributed on [0, 1). This assumption may be useful in the interpolation of time-to-event probability distribution for continuous lifetime. (a) Let F be distribution of T. Define F(x + s) = P(Tx≤s), f(x+s) = F(x+ s), f(x + s) 1- F(x + s) Let qx λx+s= = P(x ≤ T < x + 1|7 ≥x). Show for s = [0, 1) that VxEN. qx 1 - sqx (b) The parameter qx is estimated from n independent samples Ti, i = 1, ... , n using 1{x
Suppose that the fractional part S = T - K, with curtate lifetime K = [7], of lifetime T is assumed to be uniformly distributed on [0, 1). This assumption may be useful in the interpolation of time-to-event probability distribution for continuous lifetime. (a) Let F be distribution of T. Define F(x + s) = P(Tx≤s), f(x+s) = F(x+ s), f(x + s) 1- F(x + s) Let qx λx+s= = P(x ≤ T < x + 1|7 ≥x). Show for s = [0, 1) that VxEN. qx 1 - sqx (b) The parameter qx is estimated from n independent samples Ti, i = 1, ... , n using 1{x
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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Please do the following questions with handwritten working out
Transcribed Image Text:Age(x)
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1
2345
2
3
4
5
6
7
8
9
10
11
12
13
14
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Age(x)
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15
103 0 16
103
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17
100 1
18
99 1
19
98 3
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60 4
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49
39
6
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30 10
434
20
21
1 22
92 2
23
90 4
24
86 4
25
82
26
79
27
75
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28
72 8
29
1 1
Table 1: Table of mortality rates.
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![Suppose that the fractional part S = T - K, with curtate lifetime K = [T], of lifetime T
is assumed to be uniformly distributed on [0, 1). This assumption may be useful in the
interpolation of time-to-event probability distribution for continuous lifetime.
(a) Let F be distribution of T. Define F(x + s) = P(Tx ≤ s), ƒ(x + s) = ¼F(x + s),
f(x + s)
1- F(x + s)
Let qx = P(x ≤ T < x + 1|T ≥ x). Show for s = [0, 1) that
qx
Vx € N.
1 - sqx
λx+s
λx+s
=
=
(b) The parameter qx is estimated from n independent samples T;, i = 1, ..., n using
1{x<T;<x+1}
n
i=1
n
Σ 1{Tzx}
i=1
Show that is a consistent estimator of qx.
(c) Using estimator of qx in 1(b), derive an estimate for + in terms of dx and lx.
(d) Deduce from equation (1) the central exposed to risk E
aged x.
(e) Use Table 1
and calculate the estimates x,x+
Ix
and Ex](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2F6e3d27a3-0cec-42a7-bd8f-013026e8f62d%2Ff99vik_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that the fractional part S = T - K, with curtate lifetime K = [T], of lifetime T
is assumed to be uniformly distributed on [0, 1). This assumption may be useful in the
interpolation of time-to-event probability distribution for continuous lifetime.
(a) Let F be distribution of T. Define F(x + s) = P(Tx ≤ s), ƒ(x + s) = ¼F(x + s),
f(x + s)
1- F(x + s)
Let qx = P(x ≤ T < x + 1|T ≥ x). Show for s = [0, 1) that
qx
Vx € N.
1 - sqx
λx+s
λx+s
=
=
(b) The parameter qx is estimated from n independent samples T;, i = 1, ..., n using
1{x<T;<x+1}
n
i=1
n
Σ 1{Tzx}
i=1
Show that is a consistent estimator of qx.
(c) Using estimator of qx in 1(b), derive an estimate for + in terms of dx and lx.
(d) Deduce from equation (1) the central exposed to risk E
aged x.
(e) Use Table 1
and calculate the estimates x,x+
Ix
and Ex
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