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STAT 416 Stochastic Processes for Actuaries – Term 122
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
DEPARTMENT OF MATHEMATICS & STATISTICS
DHAHRAN, SAUDI ARABIA
STAT 416: Stochastic Processes for Actuaries
Semester 122
Major Exam One
Sunday, March 03, 2013
Allowed time 2 hours
:
Instructor
Adnan Jabbar
Name:
Student ID#:
Serial #:
Directions:
1) You must show all your work to obtain full credit for questions two and three.
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
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Marks
obtained
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STAT 416 Stochastic Processes for Actuaries – Term 122
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Question one (5).
Suppose an urn contains seven black balls and five white balls. We draw two balls from the urn without
replacement. Assuming that each ball in the urn is equal likely to be drawn, what is the probability that both
drawn balls are black?
Question two (5).
Suppose X has the following probability mass functions:
P (0) = 0.2, p(1) = 0.5, p(2) = 0.3
Calculate E[X2].
STAT 416 Stochastic Processes for Actuaries – Term 122
Question three (6+6).
Suppose we know that the number of items produced in a factory during a week is a random variable with
mean 500.
(a) What can be said about the probability that this week’ s production will be at least 1000?
(b) If the variance of a week’s production is known to equal 100, then what can be said about the
probability that this week’s production will be between 400 and 600?
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STAT 416 Stochastic Processes for Actuaries – Term 122
Question four (8).
A minor is trapped in a mine containing three doors. The first door leads to a tunnel that takes him to safety
after two hours of travel. The second door leads to a tunnel that returns him to the mine after three hours of
travel. The third door leads to a tunnel that returns him to the mine after five hours of travel. Assuming that
the miner is at all times equally likely to choose any one of the doors, what is the expected length of time
until the miner reaches safely?
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STAT 416 Stochastic Processes for Actuaries – Term 122
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Question five (8).
An insurance company supposes that the number of accidents that each of its policyholders will have in a
year is Poisson distributed, with the mean of the Poisson depending on the policyholder. If the Poisson mean
of a randomly chosen policyholder has a gamma distribution with density function
g (λ) = λ e-λ , λ ≥ 0
What is the probability that a randomly chosen policyholder has exactly n accidents next year?
STAT 416 Stochastic Processes for Actuaries – Term 122
Question six (4).
If immigrants to area A arrive at a Poisson rate of ten per week, and if each immigrant is of Irish descent
with probability 1/12, then what is the probability that no people of Irish descent will immigrate to area A
during the month of February?
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STAT 416 Stochastic Processes for Actuaries – Term 122
Question seven (8).
The dollar amount of damage involved in an automobile accident is an exponential random variable with
mean 1000. Of this, the insurance company only pays that amount exceeding 400. Find the expected value
and the standard deviation of the amount the insurance company pays per accident.
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