### ACTS 4301 Instructor: Natalia A. Humphreys HOMEWORK 2 Lesson

```ACTS 4301
Instructor: Natalia A. Humphreys
HOMEWORK 2
Lesson 2. Survival Distributions: Probability Functions.
Lesson 3. Survival Distributions: Force of Mortality.
Due: January 29, 2015 (Thurs)
Sufficient work must be shown to get credit for a correct answer. Partial credit may be given for
incorrect answers which have some positive work.
Problem 1
You are given the following survival function:
2500−x2
2500
S(x) =
0
0 ≤ x ≤ 50
x > 50
Calculate q43
(A) 0.8664
(B) 0.1336
(C) 0.1543
(D) 0.0471
(E) 0.0449
Problem 2
You are given the following mortality table:
x
qx
21
lx
dx
20,000 1100
22
23
17,450
24 0.0600
25 0.0680 13,200
Determine the probability that a life age 22 will die within two years.
Problem 3
You are given:
(i) The probability that a person age
(ii) The probability that a person age
(iii) The probability that a person age
Calculate the probability that a person
40 is alive at age 45 is 0.8.
45 is not alive at age 50 is 0.11.
40 is alive at age 55 is 0.57.
age 45 dies between ages 50 and 55.
Problem 4
The force of mortality is
µx =
Calculate 9 p43 .
1
90 − x
ACTS 4301. SP 2015. HOMEWORK 2.
Problem 5
Given that the force of mortality is µx = 3x2 , determine the cumulative distribution function for the
random variable time until death, F (x), the density function for that random variable, f (x), and the
survival function s(x).
Problem 6
The force of mortality for Mary is µx = 3. The force of mortality for Sarah is µx = kx4 . Determine
k for which 6 p13 is the same for Mary and Sarah.
(A) 4.2759 · 10−5
(B) 1.1574 · 10−2
(C) 2.3148 · 10−3
(D) 2.8884 · 10−4
(E) 1.0504 · 10−4
Problem 7
For a standard life, 5 p35 = 0.97. Since April, age 35, is recovering from an accident, she is subject
to extra mortality. Therefore, the µx applying to April is increased for x between 35 and 40. The
increase over the µx for a standard life is 0.004 at x=35, decreasing in a straight line to 0 at age 40.
Calculate 5 p35 for April.
Problem 8
You are given
µx =
2x
, 0 ≤ x ≤ 40
(1600 − x2 )
Determine the probability that (x) dies within a year.
Problem 9
A mortality table has a force of mortality µx+t and mortality rate qx . A second mortality table has a
force of mortality µ∗x+t and mortality rate qx∗ . You are given µ∗x+t = 0.25µx+t for 0 ≤ t ≤ 1. Calculate
qx∗ in terms of qx .
Problem 10
√
The force of mortality is µx = 0.005 3 x. Calculate the probability of someone age 18 surviving 40
years and then dying in the next 12 years.
Copyright ©Natalia A. Humphreys, 2015
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