17. The length of time, in years, that a person will remember an actuarial statistic is modeled by
an exponential distribution with mean 1
Y . In a certain population, Y has a gamma
distribution with 2 α θ = = .
Calculate the probability that a person drawn at random from this population will remember
an actuarial statistic less than 1
2 year.
(A) 0.125
(B) 0.250
(C) 0.500
(D) 0.750
(E) 0.875
Exam M: Fall 2005 -18- GO ON TO NEXT PAGE
18. In a CCRC, residents start each month in one of the following three states: Independent
Living (State #1), Temporarily in a Health Center (State #2) or Permanently in a Health
Center (State #3). Transitions between states occur at the end of the month.
If a resident receives physical therapy, the number of sessions that the resident receives in a
month has a geometric distribution with a mean which depends on the state in which the
resident begins the month. The numbers of sessions received are independent. The number
in each state at the beginning of a given month, the probability of needing physical therapy in
the month, and the mean number of sessions received for residents receiving therapy are
displayed in the following table:
State # Number in
state
Probability of
needing therapy
Mean number
of visits
1 400 0.2 2
2 300 0.5 15
3 200 0.3 9
Using the normal approximation for the aggregate distribution, calculate the probability that
more than 3000 physical therapy sessions will be required for the given month.
(A) 0.21
(B) 0.27
(C) 0.34
(D) 0.42
(E) 0.50
Exam M: Fall 2005 -19- GO ON TO NEXT PAGE
19. In a given week, the number of projects that require you to work overtime has a geometric
distribution with 2 β= . For each project, the distribution of the number of overtime hours in
the week is the following:
x ( ) f x
5 0.2
10 0.3
20 0.5
The number of projects and number of overtime hours are independent. You will get paid for
overtime hours in excess of 15 hours in the week.
Calculate the expected number of overtime hours for which you will get paid in the week.
(A) 18.5
(B) 18.8
(C) 22.1
(D) 26.2
(E) 28.0
Exam M: Fall 2005 -20- GO ON TO NEXT PAGE
20. For a group of lives age x, you are given:
(i) Each member of the group has a constant force of mortality that is drawn from the
uniform distribution on [ ] 0.01, 0.02 .
(ii) 0.01 δ=
For a member selected at random from this group, calculate the actuarial present value of a
continuous lifetime annuity of 1 per year.
(A) 40.0
(B) 40.5
(C) 41.1
(D) 41.7
(E) 42.3
Exam M: Fall 2005 -21- GO ON TO NEXT PAGE
21. For a population whose mortality follows DeMoivre’s law, you are given:
(i) 40:40 60:60 3 e e = 􀁄 􀁄
(ii) 20:20 60:60 e ke = 􀁄 􀁄
Calculate k.
(A) 3.0
(B) 3.5
(C) 4.0
(D) 4.5
(E) 5.0
Exam M: Fall 2005 -22- GO ON TO NEXT PAGE
22. For an insurance on (x) and (y):
(i) Upon the first death, the survivor receives the single benefit premium for a whole life
insurance of 10,000 payable at the moment of death of the survivor.
(ii) ( ) ( ) 0.06 x y t t µ µ = = while both are alive.
(iii) ( ) 0.12 x y t µ =
(iv) After the first death, ( ) 0.10 t µ = for the survivor.
(v) 0.04 δ=
Calculate the actuarial present value of this insurance on (x) and (y).
(A) 4500
(B) 5400
(C) 6000
(D) 7100
(E) 7500
Exam M: Fall 2005 -23- GO ON TO NEXT PAGE