Notes-Ch10-11-331W09 - Life Contingencies 1 ACTSC 331...

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

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
.