5. A certain species of flower has three states: sustainable, endangered and extinct. Transitions
between states are modeled as a non-homogeneous Markov chain with transition matrices i Q
as follows:
1
Endangered Sustainable Extinct
Sustainable 0.85 0.15 0
0 0.7 0.3 Endangered
Extinct 0 0 1
Q
⎛ ⎞
⎜ ⎟
⎜ ⎟ = ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ ⎠
2
0.9 0.1 0
0.1 0.7 0.2
0 0 1
Q
⎛ ⎞
⎜ ⎟ =⎜ ⎟
⎜ ⎟
⎝ ⎠
3
0.95 0.05 0
0.2 0.7 0.1
0 0 1
Q
⎛ ⎞
⎜ ⎟ =⎜ ⎟
⎜ ⎟
⎝ ⎠
0.95 0.05 0
0.5 0.5 0 , 4,5,...
0 0 1
iQ i
⎛ ⎞
⎜ ⎟ = = ⎜ ⎟
⎜ ⎟
⎝ ⎠
Calculate the probability that a species endangered at the start of year 1 will ever become
extinct.
(A) 0.45
(B) 0.47
(C) 0.49
(D) 0.51
(E) 0.53
Exam M: Fall 2005 -6- GO ON TO NEXT PAGE
6. For a special 3-year term insurance:
(i) Insureds may be in one of three states at the beginning of each year: active, disabled,
or dead. All insureds are initially active. The annual transition probabilities are as
follows:
Active Disabled Dead
Active 0.8 0.1 0.1
Disabled 0.1 0.7 0.2
Dead 0.0 0.0 1.0
(ii) A 100,000 benefit is payable at the end of the year of death whether the insured was
active or disabled.
(iii) Premiums are paid at the beginning of each year when active. Insureds do not pay
any annual premiums when they are disabled.
(iv) d = 0.10
Calculate the level annual benefit premium for this insurance.
(A) 9,000
(B) 10,700
(C) 11,800
(D) 13,200
(E) 20,800
Exam M: Fall 2005 -7- GO ON TO NEXT PAGE
7. Customers arrive at a bank according to a Poisson process at the rate of 100 per hour. 20%
of them make only a deposit, 30% make only a withdrawal and the remaining 50% are there
only to complain. Deposit amounts are distributed with mean 8000 and standard deviation
1000. Withdrawal amounts have mean 5000 and standard deviation 2000.
The number of customers and their activities are mutually independent.
Using the normal approximation, calculate the probability that for an 8-hour day the total
withdrawals of the bank will exceed the total deposits.
(A) 0.27
(B) 0.30
(C) 0.33
(D) 0.36
(E) 0.39
Exam M: Fall 2005 -8- GO ON TO NEXT PAGE
8. A Mars probe has two batteries. Once a battery is activated, its future lifetime is exponential
with mean 1 year.
The first battery is activated when the probe lands on Mars. The second battery is activated
when the first fails.
Battery lifetimes after activation are independent.
The probe transmits data until both batteries have failed.
Calculate the probability that the probe is transmitting data three years after landing.
(A) 0.05
(B) 0.10
(C) 0.15
(D) 0.20
(E) 0.25
Exam M: Fall 2005 -9- GO ON TO NEXT PAGE