Krzys’ Ostaszewski: http://www.krzysio.net
Author of the “Been There Done That!” manual for Course P/1
http://smartURL.it/krzysioP (paper) or http://smartURL.it/krzysioPe (electronic)
Instructor of online P/1 seminar: http://smartURL.it/onlineactuary
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P Sample Exam Questions, Problem No. 148, also Dr. Ostaszewski’s online exercise
posted March 6, 2010
The number of hurricanes that will hit a certain house in the next ten years is Poisson
distributed with mean 4. Each hurricane results in a loss that is exponentially distributed
with mean 1000. Losses are mutually independent and independent of the number of
hurricanes. Calculate the variance of the total loss due to hurricanes hitting this house in
the next ten years.
A. 4,000,000
B. 4,004,000
C. 8,000,000
D. 16,000,000
E. 20,000,000
Solution.
Let N denote the number of hurricanes. Then N has a Poisson distribution with mean and
variance 4. Let Xi denote the loss caused by the i-th hurricane. Then Xi has an
exponential distribution with mean 1000 and variance 1000 2 = 1000000. Let X denote the
total loss due to N hurricanes. Note that X1 , X2 , … and X N are independent, and
therefore
Var ( X ) = Var E ( X N ) + E Var ( X N ) = Var ( NE ( X1 )) + E ( NVar ( X1 )) =
(
)
(
)
= Var (1000N ) + E (1000000N ) = 1000 2 ⋅ Var ( N ) + 1000000E ( N ) =
Answer C.
= 1000 2 ⋅ 4 + 1000000 ⋅ 4 = 8000000.
© Copyright 2010 by Krzysztof Ostaszewski.
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Exercises from the past actuarial examinations are copyrighted by the Society of
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