Practice Questions 6 – ACTSC 331, FALL 2011
1. The surplus process of an insurer is
U
t
=
u
+ 1
.
2
t

∑
N
t
i
=1
X
i
, t
≥
0, where
u
≥
0 is the
initial surplus; the claim number process
{
N
t
, t
≥
0
}
is a Poisson process with rate
1; and the claim sizes
{
X
1
, X
2
, ...
}
,
independent of
{
N
t
, t
≥
0
}
, are independent and
identically distributed exponential random variables with mean 1.
(a) Calculate the probability that the number of claims is one in the first three months
and two in the first five months.
(b) Given that there are three claims in the first half year, what is the probability
that there is no claim in the first month?
(c) Given that there are five claims in the first year, what is the probability that
exactly two of the five are each bigger than one?
(d) Calculate the probability that there are exactly two claims in the first year, which
are each bigger than one.
(e) Given that the 6th claim occurs at the end of twenty months, what is the expected
time of the 2nd claim?
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 Fall '09
 laundriualt
 Poisson Distribution, Probability theory, Exponential distribution, Yi, initial surplus

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