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Review Notes for Loss Models 1  ACTSC 431/831, FALL 2008
Part 3 – Frequency Models
1.
Frequency models
are used to model the number of events or claims.
2.
A counting random variable
N
is a nonnegative integervalued random variable
with pf
p
k
=
P
{
N
=
k
}
, k
= 0
,
1
,
2
,...
and the distribution of a counting random
variable is called a
counting distribution
.
(a) The pgf of a counting random variable is given by
P
N
(
z
) =
E
(
z
N
) =
∞
X
n
=0
p
n
z
n
=
p
0
+
p
1
z
+
p
2
z
2
+
· · ·
,
in particular
p
0
=
P
N
(0),
E
(
N
) =
P
0
N
(1)
, E
(
N
(
N

1)) =
P
00
N
(1)
,
and
E
(
N
2
) =
P
00
N
(1) +
P
0
N
(1)
.
3.
Three important counting distributions:
(a) Poisson distribution with
E
(
N
) =
V ar
(
N
).
(b) Binomial distribution with
E
(
N
)
> V ar
(
N
).
(c) Negative binomial distribution with
E
(
N
)
< V ar
(
N
)
.
i. A negative binomial distribution
NB
(1
,β
) is called a
geometric distribu
tion
with pf
Pr
{
N
=
k
}
=
1
1 +
β
±
β
1 +
β
!
k
, k
= 0
,
1
,
2
,..., β >
0
.
Alternately, a counting random variable
N
is said to be a geometric distrib
ution if the pf of
N
has the following form:
Pr
{
N
=
k
}
=
θ
(1

θ
)
k
, k
= 0
,
1
,
2
,...,
0
< θ <
1
with
E
(
N
) =
1

θ
θ
and
V ar
(
N
) =
1

θ
θ
2
.
ii.
Theorem (The memoryless property of a geometric distribution):
If
N
has a geometric distribution
NB
(1
,β
), then for any
n
= 0
,
1
,
2
,...,
N

n

N
≥
n
has the same geometric distribution as that of
N
, namely
Pr
{
N

n
=
k

N
≥
n
}
= Pr
{
N
=
k
}
for
k
= 0
,
1
,
2
,....
1
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Relationships between
P
(
λ
)
,
NB
(
r,β
)
, and
b
(
n,p
):
(a) A negative binomial distribution
NB
(
r,β
) is the mixture of a Poisson distribution
P
(
λ
) and a gamma distribution
G
(
r,β
), i.e. if
X

Λ =
λ
has a Poisson distribution
P
(
λ
) and Λ has a gamma distribution
G
(
r,β
), then
X
has a negative binomial
distribution
NB
(
r,β
).
(b) A Poisson distribution
P
(
λ
) is the limit of a sequence of binomial distributions
{
b
(
n,
λ
n
)
}
, namely if
X
n
has a binomial distribution
b
(
n,
λ
n
), then
lim
n
→∞
Pr
{
X
n
=
k
}
=
e

λ
λ
k
k
!
.
5.
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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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