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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Actuarial Science Program
DEPARTMENT OF MATHEMATICS
Math 476 / 567 Prof. Rick Gorvett
Actuarial Risk Theory Fall, 2008
Homework Assignment # 3 (max. points = 10)
Due at the beginning of class on Thursday, September 18, 2008
You are encouraged to work on these problems in groups of no more than 3 or 4. However, each
student must hand in her/his own answer sheet. Please show your work – enough to show that
you understand how to do the problem – and circle your final answer. Full credit can only be
given if the answer and approach are appropriate. Please provide answers to two decimal places.
Note: When calculating a percentage, please provide answers either as a proportion to
four decimal places (e.g., 0.xxxx), or as a percentage to two decimal places (e.g., xx.xx%).
(1) Grateful students arrive at a professor’s office, during his daily two-hour office hour
period, at a Poisson rate of 12 per hour. Each student offers a box of cookies as a gift to
the chubby but lovable fellow. The number of cookies in each student’s gift box follows
a continuous uniform distribution, with between 0 and 4 cookies per box. Assume that
the professor eats the cookies in each box immediately upon receipt (what did you
expect? There’s nothing worse than a stale cookie…). The professor will explode (not a
pretty sight, by the way) if he eats more than 60 cookies in one day. Approximate the
probability that the professor will explode (due to cookie over-eating) on any given day
during his office hours.
(2) Same problem as (1) above, the only difference being that each box contains either 0
cookies or 4 cookies, with equal probabilities. (By the way, those students disappointing
the professor by having 0 cookies in their box will be dealt with most seriously…)
(3) An insurance company receives claims according to a Poisson process. The claims are of
two types: auto liability (AL) with probability 0.75, and workers compensation (WC)
with probability 0.25. For a given day, and given that the company receives at least four
claims on that day, what is the probability that the company receives the second WC
claim before it receives the third AL claim?
(4) Suppose that X follows an arithmetic Brownian motion process, according to the formula:
dX = 5dt + 8dz
If X(2) = 125, find a 95% confidence interval around X(5).
