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-STAT 416 Stochastic Processes for Actuaries – Term 122
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KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
DEPARTMENT OF MATHEMATICS & STATISTICS
DHAHRAN, SAUDI ARABIA
STAT 416: Stochastic Processes for Actuaries
Semester 122
Major Exam Final
Sunday, May 26, 2013
Allowed time 3 hours
:
Instructor
Adnan Jabbar
Name:
Student ID#:
Serial #:
Directions:
1) You must show all your work to obtain full credit for all questions.
2) You are allowed to use electronic calculators and other reasonable writing accessories that
help write the exam. Try to define events, formulate problem and solve.
3) Do not keep your mobile with you during the exam, turn off your mobile and leave it aside
Question No
Full marks
1
7
2
9
3
7
4
11
5
9
6
14
7
10
8
7
9
7
10
10
11
9
Marks
obtained
-STAT 416 Stochastic Processes for Actuaries – Term 122
Question one (7).
Suppose that an airplane engine will fail, when in flight, with probability (1- p) independently from engine
to engine; suppose that the airplane will make a successful flight if at least 50% of its engines remain
operative. For what values of p is a four-engine plane preferable to a two-engine plane?
2
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question two (3 + 2 + 4).
It is necessary to simulate samples from a distribution with density function
f (x) = 6x ( 1 – x)
0 < x < 1.
(a) Use the acceptance-rejection technique to construct a complete algorithm for generating samples from f
by first generating samples from the distribution with density h (x) = 2(1 –x)
(b) Calculate how many samples from h would on average be needed to generate one realisation from f.
(c) Explain whether the acceptance-rejection method in (i) would be more efficient if the uniform
distribution were to be used instead.
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question three (1+6)
Three white and three black balls are distributed in two urns in such a way that each contains three balls. We
say that the system is in state i , i = 0,1,2,3, if the first urn contains i white balls. At each step, we draw one
ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball
from the second urn. Let Xn denote the state of system after the nth step. Explain why { Xn , n = 0, 1, 2, ….}
is a markov chain and calculate its transition probability matrix.
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question four (5 + 6).
(a) Suppose you arrive at a post office having two clerks at a moment when both are busy but there is no one
else waiting in line. You will enter service when either clerk becomes free. If service times for clerk i are
exponential with rate λi, i = 1,2, find E[T], where T is the amount of time that you spend in the post office.
(b) Suppose that a one-selled organism can be in one of two states-either A or B. An individual in state A
will change to state B at an exponential rate α; an individual in state B divides into two new individuals of
type A at an exponential rate β. Define an appropriate continuous –time Markov chain for a population of
such organisms and determine the appropriate parameters for his model.
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question five (5 + 4)
Let S(t) denote the price of a security at time t. A popular model for the process { S( t), t > 0 } suppose that
the price remains unchanged until a “stock” occurs, at which time the price is multiplied by a random factor.
If we let N(t) denote the number of stocks by time t, and let Xi denote the ith multiplicative factor, then this
model supposes that
N(t)
S(t) = S(0) Π Xi
i=1
N(t)
Where Π Xi is equal to 1 when N(t) =0.
i=1
Suppose that the Xi are independent exponential random variables with rate μ; that {N (t), t > 0} is a poisson
process with rate λ; that {N (t), t > 0}is independent of the Xi; and that S(0) = s.
(a) Find E[S (t)]
(b) Find E [S2 (t)].
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question six (4+4+2+4).
An insurer operates a simple no claims discount system with 5 levels: 0%, 20%, 40%, 50% and 60%.
The rules for moving between levels are:
An introductory discount of 20% is available to new customers.
If no claims are made during a year the policyholder moves up to the next discount level or remains at the
maximum level.
If one or more claims are made during the year, a policyholder at the 50% or 60% discount level moves to
the 20% level and a policyholder at 0%, 20% or 40% moves to or remains at the 0% level.
The full annual premium is £600.
When an accident occurs, the distribution of loss is exponential with mean £1,750. A policyholder will only
claim if the loss is greater than the extra premiums over the next four years, assuming no further accidents
occur.
(a) For each discount level, calculate the smallest cost for which a policyholder will make a claim.
-STAT 416 Stochastic Processes for Actuaries – Term 122
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(b) For each discount level, calculate the probability of a claim being made in the event of an accident
occurring.
(c) Comment on the results of (b).
-STAT 416 Stochastic Processes for Actuaries – Term 122
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(d) Currently, equal numbers of customers are in each discount level and the probability of a policyholder
not having an accident each year is 0.9. Calculate the expected proportions in each discount level next year.
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question seven (2 +2 + 6).
An insurer has an initial surplus of U. Claims up to time t are denoted by S(t). Annual premium income is
received continuously at a rate of c per unit time.
(a) Explain what is meant by the insurer’s surplus process U(t).
(b) Define carefully each of the following probabilities:
i.
ψ ( U, t)
ii.
ψt ( U, t)
(c) Explain, for each of the following pairs of expressions, whether one of each pair is certainly greater
than the other, or whether it is not possible to reach a conclusion.
i.
ψ (10, 2) and ψ (20, 1)
ii.
ψ (10,2) and ψ (5, 1)
iii. ψ0.5(10,2) and ψ0.25 (10,2)
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question eight (7)
Assume that a stock price S follows geometric Brownian motion. Then prove that the stock price has a
lognormally distributed.
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question nine (1 + 6)
A sequence of pseudo-random numbers from a uniform distribution over the interval [0, 1] has been
generated by a computer.
(a) Explain the advantage of using pseudo-random numbers rather than generating a new set of random
numbers each time.
(b) Use examples to explain how a sequence of pseudo-random numbers can be used to simulate
observations from:
i.
a continuous distribution
ii.
a discrete distribution
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question ten (6 + 4)
(a) Let n be an integer and suppose that X1, X2, , Xn are independent random variables each having an
exponential distribution with parameter . Show that Z = X1 + + Xn has a Gamma distribution with
parameters n and λ .
(b) By using this result, generate a random sample from a Gamma distribution with mean 30 and variance
300 using the 5 digit pseudo-random numbers.
63293
43937
08513
-STAT 416 Stochastic Processes for Actuaries – Term 122
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Question eleven (8)
An insurance company has a portfolio of two-year policies. Aggregate annual claims from the portfolio
follow an exponential distribution with mean 10 (independently from year to year). Annual premiums of 15
are payable at the start of each year. The insurer checks for ruin only at the end of each year. The insurer
starts with no capital. Calculate the probability that the insurer is not ruined by the end of the second year.