Midterm test
MATH 3280 3.00
October 20th, 2014
Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This exam is a ‘closed book’ exam, duration 80 minutes.
3. Only non-programmable calculators are permitted.
4. The text has ten pages, and it contains eleven questions. Read the
questions carefully. Fill in answers in designated spaces. Your work
must justify the answer you give. Answers without supporting work
will not be given credit.
5. Hand in your examination answer booklets.
Useful formulas:
1.) For t ∈ [0, 1), and non-negative integer k, we have that
U DD
P[T (u) ≥ k + t] = (1 − t)P[T (u) ≥ k] + tP[T (u) ≥ k + 1].
2.) For ∆f (k) := f (k + 1) − f (k), k ≥ 0, it holds that
n
X
k=m
n
X
n+1
f (k)∆g(k) = f (k)g(k)|m −
g(k + 1)∆f (k).
k=m
GOOD LUCK!
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MATH 3280 3.00
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Question 1 Let T (u) be an absolutely continuous r.v. representing the future life time of a life
status (u) (note that (u) can be any life status we have seen. e.g., (x), (n ), etc).
1.1 Recall that t pu := P[T (u) ≥ t] with t ≥ 0, show that this function is leftcontinuous. Give an example of (u), such that ∞ pu 6= 0.
1
1.2 Specialize your life status to (x : n ), derive the full expectancy of life.
1
1.3 Let q ∈ (0, 1), then formulate F 1−1 (q) := inf{x : P[T (x : n ) ≥ x] ≤ (1 − q)}.
x:n
1.4 Assume that the UDD assumption holds for (x). Verify whether the UDD as1
sumption holds for (u) = (x : n ).
1.5 Let µ(x) = Bcx , B > 0, c > 1 with x ≥ 0. Show that the function lx µ(x) has its
maximum at an age x0 , such that µ(x0 ) = log c, c > 1.
Cont.
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Question 2 Let K(u) denote the curtuate future life time r.v. (note that (u) can be any life status
we have seen. e.g., (x), (n ), n ∈ N, etc). Also, let T (u) be as in Questions 1.
2.1 Define K(n ) in terms of T (n ).
1
2.2 Let (u) = (x : n ), derive k px:n
, for k = 0, 1, . . ..
1
1
2.3 Develop the probability mass function of the r.v. K(x : n ), i.e., formulate the
1
probability P[K(x : n ) = k], for all k = 0, 1, . . ..
2.4 Consider the special ratio below
R1
tt px µ(x + t)dt
a(x) = R01
= E[T (x)| T (x) < 1],
p
µ(x
+
t)dt
t
x
0
where x ≥ 0. Give a short (one sentence only) interpretation of a(x), x ≥ 0.
Show that under the CFM assumption on (x),
CF M
a(x) =
1 px
− , x ≥ 0.
µ qx
2.5 You are given that the future life times of two individuals (30) and (34) are
independent and such that:
x
30
31
32
33
34
35
36
37
qx
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Calculate 2| q30:34 , i.e., the probability that the first death occurs during the third
year.
.
Cont.
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Question 3 The life status (50) is subject to an extra hazard during the year [50, 51). If the
standard probability of death from age 50 to age 51 is 0.006, and if the extra risk
may be expressed by an addition to the standard force of mortality, that decreases
uniformly from 0.03 at the beginning of the year to zero at the end of the year,
calculate the probability that (50) will survive to age 51.
.
Cont.
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The End.
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