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SOLUTIONS TO ASSIGNMENT 4 – ACTSC 431/831, FALL 2008
1. The conditional distribution of
X
, given Θ =
θ
, is a Poisson distribution with mean
λθ
. The pgf of
X
is given by
P
X
(
z
) =
M
Θ
(
λ
(
z

1))
,
where
M
Θ
(
t
) is the mgf of Θ.
Since Θ is inﬁnitely divisible, for any
n
= 1
,
2
,...,
there exists a mgf
M
n
(
t
) so that
M
Θ
(
t
) = [
M
n
(
t
)]
n
.
Thus,
P
X
(
z
) = [
M
n
(
λ
(
z

1))]
n
.
Since
M
n
(
λ
(
z

1)) is still a pgf
that is the pgf of a mixed Poisson distribution, in which the mixing distribution has
the mgf
M
n
(
t
). Hence,
X
is inﬁnitely divisible.
2. (a) Let
v
= Pr
{
X
j
≤
50
}
and let
I
j
= 1 if
X
j
≤
50 and 0 if
X
j
>
50. Note that
N
P
*
=
I
1
+
...
+
I
N
L
. If
N
L
is inﬁnitely divisible, then for any
n
= 1
,
2
,...,
there
exists a pgf
P
n
(
z
) so that
P
N
L
(
z
) = [
P
n
(
z
)]
n
.
Thus, the pgf of
N
P
*
is
P
N
P
*
(
z
) =
P
N
L
(1 +
v
(
z

1)) = [
P
n
(1 +
v
(
z

1))]
n
,
which means that
N
P
*
is also inﬁnitely divisible since
P
n
(1+
v
(
z

1)) is still a pgf
that is the pgf of a compound frequency model, in which
P
n
(
z
) is the pgf of the
primary distribution and the secondary distribution is the Bernoulli distribution
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This note was uploaded on 02/02/2010 for the course ACTSC 331 taught by Professor David during the Fall '09 term at Waterloo.
 Fall '09
 david

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