Life Contingencies 1  ACTSC 331, Winter 2009
Review Notes for Chapters 10 and 11
1.
The multiple decrement model:
Assume that life (
x
) is a member of a group. Let
T
(
x
) =
T
denote the time of decrement at which life (
x
) leaves the group.
T
is a
nonnegative continuous random variable. Further, assume that there are
m
causes of
decrement. Let
J
(
x
) =
J
be a discrete random variable and let (
J
=
j
) denote cause
j, j
= 1
,
2
, ..., m
. Then
J
takes only
m
possible values of 1
,
2
, ..., m.
2.
Distributions of
T
and
J
:
(a) The marginal p.d.f. of
T
is denoted by
f
T
(
t
)
, t
≥
0 with
R
∞
0
f
T
(
t
)
dt
= 1
.
(b) The marginal p.f.
of
J
is denoted by
f
J
(
j
) = Pr
{
J
=
j
}
, j
= 1
,
2
, ..., m
with
∑
m
j
=1
f
J
(
j
) = 1.
(c) The joint density function of
T
and
J
is denoted by
f
T,J
(
t, j
)
,
t
≥
0
,
j
=
1
,
2
, ..., m.
(d) Relationships between
f
T
(
t
)
, f
J
(
j
)
,
and
f
T,J
(
t, j
):
f
T
(
t
) =
m
X
j
=1
f
T,J
(
t, j
)
and
f
J
(
j
) =
Z
∞
0
f
T,J
(
t, j
)
dt.
3.
Probabilities of decrements:
(a) The probability of decrement between times
a
and
b
due to cause
j
is
Pr
{
a < T
≤
b, J
=
j
}
=
Z
b
a
f
T,J
(
t, j
)
dt.
(b) The probability of decrement between times
a
and
b
due to all causes is
Pr
{
a < T
≤
b
}
=
m
X
j
=1
Pr
{
a < T
≤
b, J
=
j
}
=
m
X
j
=1
Z
b
a
f
T,J
(
t, j
)
dt.
(c) The probability of decrement before time
t
due to cause
j
is denoted by
t
q
(
j
)
x
and
t
q
(
j
)
x
= Pr
{
0
≤
T
≤
t, J
=
j
}
=
Z
t
0
f
T,J
(
s, j
)
ds
with
d
dt
t
q
(
j
)
x
=
f
T,J
(
t, j
)
.
Note:
t
q
(
j
)
x
is not a distribution function.
(d) The probability of decrement at any time in the future due to cause
j
is denoted
by
∞
q
(
j
)
x
and
∞
q
(
j
)
x
= Pr
{
0
≤
T <
∞
, J
=
j
}
=
f
J
(
j
) =
Z
∞
0
f
T,J
(
t, j
)
dt.
1
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4.
Distributions of
T
:
(a) The distribution function of
T
is denoted by
t
q
(
τ
)
x
and
t
q
(
τ
)
x
= Pr
{
T
≤
t
}
=
F
T
(
t
) =
Z
t
0
f
T
(
s
)
ds,
which is the probability of decrement before time
t
due to all causes.
(b) The survival function of
T
is denoted by
t
p
(
τ
)
x
and
t
p
(
τ
)
x
= 1

t
q
(
τ
)
x
= Pr
{
T > t
}
=
Z
∞
t
f
T
(
s
)
ds,
which is the probability that life (
x
) is still in the group at age
x
+
t
.
(c) The total force of decrement is denoted by
μ
(
τ
)
x
(
t
) and
μ
(
τ
)
x
(
t
) =
f
T
(
t
)
1

F
T
(
t
)
=
d
dt
t
q
(
τ
)
x
t
p
(
τ
)
x
.
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 Spring '09
 david

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