STAT 4004 Assignment 2 Due: October 19, 2020
1. Suppose the survivor function of the newborn lifetime T0 is given by
s(x) = exp
(
−
(
x
θ
)τ)
, x > 0
Find
(a) the density function of T0
(b) the hazard function of T0
(c) the probability that T0 is alive at 5 given T0 > 2
2. Suppose the survival function of the curtate lifetime of a newborn baby is given
by
s(x) =
100− x
100
, x = 0, 1, . . . , 100.
(a) the c.d.f. of K30
(b) the survival function of K30
(c) the probability mass function of K30
3. If lx = k(100− x)2 for 0 ≤ x ≤ 100, find
(a) 3p60
(b) 10|5q50
4. If K is the curtate future lifetime of (96), calculate Var(K), given the following
life table:
x 96 97 98 99 100
`x 180 130 73 31 0
5. A life at age 50 is subject to an extra hazard during the year of age 50 to 51.
If the standard probability of death from age 50 to 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 0 at the end of
the year, calculate the probability that the life will survive to age 51.
6. From a life table with a one-year select period, you are given:
x `[x] d[x]
◦
e[x]
85 1000 100 5.556
86 850 100
Assume that deaths are uniformly distributed over each year of age. Calculate
◦
e[86].
7. Mr. Q Taro has only 3 hairs left on his head and he won’t be growing any more.
(a) The future mortality of each hair follows
k|qx = 0.1(k + 1), k = 0, 1, 2, 3 and x is Mr. Q Taro′s age
(b) Hair loss follows the uniform distribution of death assumption at fractional
ages.
(c) The future lifetime of 3 hairs are independent.
Calculate the probability that Mr. Q Taro is bald (has no hair left) at age
x+ 2.5.
8. Mortality for Don, aged 25, follows De Moivre’s law with ω = 100. If she
takes up hot air ballooning for the coming year, her assumed mortality will
be adjusted so that for the coming year only, she will have a constant force of
mortality of 0.1.
Calculate the decrease in the complete life expectancy for Don if she takes up
hot air ballooning.
9. For a ten-year select-and-ultimate table, you are given:
(a)
l[30]+t =
√
60
9
(
1− t
100
)
, 0 ≤ t < 10
(b)
l30+t =
√
70− t
10
, 10 ≤ t ≤ 70
Calculate
◦
e[30].
10. Show that
e
x:n =
n∑
k=1
kpx
where n is an integer.

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