Practice Questions 4 – ACTSC 431/831, FALL 2011
1. Prove that if
X
has a mixed Poisson distribution and the mixing distribution is infinitely
divisible, then
X
is infinitely divisible.
2. Let
N
L
be the number of losses. The size of the
j
th loss is
X
j
. Assume that
N
L
, X
1
, X
2
, ...
are
independent and
X
1
, X
2
, ...
have the same distribution as a Pareto distribution
Pareto
(3
,
100).
A loss will result in a payment when the loss exceeds an ordinary deductible of 50. Let
N
P
be
the number of payments and
N
P
*
be the number of losses which will not result in payments.
(a) Assume that
N
L
is infinitely divisible. Show that
N
P
*
is infinitely divisible.
(b) Assume that
N
L
has the same distribution as that the compound frequency model
∑
N
i
=1
M
i
, where the primary distribution is a negative binomial
NB
(4
,
9) and the second
distribution is a Poisson distribution with mean 8.
i. Calculate the variance of the number of payments.
ii. Calculate
Cov
(
N
P
, N
P
*
).
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 Fall '09
 laundriualt
 Variance, Probability theory, Cumulative distribution function, Discrete probability distribution

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