Review Notes for Loss Models 1  ACTSC 431/831, FALL 2011
Part 2 – Severity Models
1.
Severity models
are distributions that are used to model the distribution of the
amount of a claim/loss.
2. The distribution function
F
(
y
) of a random variable
Y
is called
an
n
point mixture
distribution
if
F
(
y
) =
α
1
F
X
1
(
y
) +
· · ·
+
α
n
F
X
n
(
y
)
for all
y,
where
α
j
≥
0,
F
X
j
(
y
) is the distribution function of a random variable
X
j
for
j
=
1
, ..., n
, and
α
1
+
· · ·
+
α
n
= 1.
(a) The survival function of the
n
point mixture distribution is
S
(
y
) =
α
1
S
X
1
(
y
) +
· · ·
+
α
n
S
X
n
(
y
)
for all
y.
(b) If
F
X
j
(
y
) has pdf
f
X
j
(
y
) for
j
= 1
, ..., n
, then the pdf of the
n
point mixture
distribution is
f
(
y
) =
α
1
f
X
1
(
y
) +
· · ·
+
α
n
f
X
n
(
y
)
for all
y.
(c) The
k
th moment of the
n
point mixture distribution is
E
(
Y
k
) =
α
1
E
(
X
k
1
) +
· · ·
+
α
n
E
(
X
k
n
)
.
(d) The moment generating function (mgf) of the
n
point mixture distribution is
M
Y
(
t
) =
α
1
M
X
1
(
t
) +
· · ·
+
α
n
M
X
n
(
t
)
,
where, the moment generating function (mgf) of a random variable
X
or its
distribution is defined as
M
X
(
t
) =
E
(
e
tX
) wherever this expectation exists for
some
t >
0. The mgf of
X
has the following properties:
i.
M
0
X
(0) =
E
(
X
) and
M
00
X
(0) =
E
(
X
2
).
ii. The distribution of a random variable
X
is determined uniquely by its mo
ment generating function, namely there is a onetoone relationship between
the mgf and its distribution.
iii. If random variables
X
and
Y
are independent, then
M
X
+
Y
(
t
) =
M
X
(
t
)
M
Y
(
t
).
1
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3.
The tail probability or tail
of a random variable
X
or its distribution
F
is the
probability that
X
exceeds a positive value
t
, or
Pr
{
X > t
}
= 1

F
(
t
) =
S
(
t
)
, t >
0
.
(a) Comparison between the tails of two random variables: Suppose that
X
and
Y
have distribution functions
F
X
(
x
) and
F
Y
(
x
), respectively. Assume that
lim
x
→∞
Pr
{
X > x
}
Pr
{
Y > x
}
= lim
x
→∞
S
X
(
x
)
S
Y
(
x
)
=
c.
If
c
= 0, we say that
X
(
F
X
(
x
)) has a
lighter tail
than
Y
(
F
Y
(
x
)); if 0
< c <
∞
,
we say that
X
(
F
X
(
x
)) and
Y
(
F
Y
(
x
)) have
proportional tails
; if
c
=
∞
, we
say that
X
(
F
X
(
x
)) has a
heavier tail
than
Y
(
F
Y
(
x
)).
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 Fall '09
 laundriualt
 Probability theory, Fθ

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