Sections 6.1, 6.2, 6.3, 6.4
Math 400 - Actuarial Models
NAME:_____________________________
1. The life table below is defined by certain values of S0.
x
S0(x)
px
qx
lx
dx
____________________________________________________
0 1.000000
.6838
.3162
1000000
316228
1
.683772
.8084
.1916
683772
130986
2
.552786
.8182
.1818
552786
100509
3
.452277
.8127
.1873
452277
84733
4
.367544
.7969
.2031
367544
74651
5
.292893
.7696
.2304
292893
67490
6
.225403
7
.163340
8
.105573
9
.051317
10 .000000
(a) Complete the table.
(b) What is the value for ?
1
Sections 6.1, 6.2, 6.3, 6.4
1.-continued
(c) Find each of the following:
3q2
=
2q3
=
q3 =
2 | q3
=
2 | 4 q3
=
4 | 2 q3
=
3q7
=
3p2
=
4p5
=
p8 =
4p8
=
2
Sections 6.1, 6.2, 6.3, 6.4
2. Consider a lifetime random variable Q with p.d.f. f(q) and survival
function S(q).
(a) Use integration by parts to find a formula for E(Q) which is an
alternative to the basic definition.
The basic definition is
We shall use the integration by parts equation
with u = q and v / = f(q).
u/=
3
and v =
Sections 6.1, 6.2, 6.3, 6.4
2.-continued
(b) Use integration by parts to find a formula for E(Q 2) which is an
alternative to the basic definition.
The basic definition is
We shall use the integration by parts equation
with u = q2 and v / = f(q).
u/=
4
and v =
Sections 6.1, 6.2, 6.3, 6.4
2.-continued
Next, we shall use the integration by parts equation
with u = q and v / = S(q).
u/=
5
and v =
Sections 6.1, 6.2, 6.3, 6.4
3. Let T0 be a lifetime random variable with survival function
S0(t) = 1  (t / ) for 0  t  
where  and  are a positive constants.
(a) Find a formula for each of E(T0), E(T02), and Var(T0).
6
Sections 6.1, 6.2, 6.3, 6.4
3.-continued
(b) Let Tx be the future lifetime random variable for an entity age x. Find
a formula for E(Tx); then with  = 2, find a formula each of E(Tx2) and
Var(Tx).
7
Sections 6.1, 6.2, 6.3, 6.4
3.-continued
(c) Suppose  = 2 and  = 10. Find each of the following:
3q2
=
2q3
=
q3 =
2 | q3
=
2 | 4 q3
=
4 | 2 q3
=
3q7
=
3p2
=
4p5
=
p8 =
4p8
=
8
Sections 6.1, 6.2, 6.3, 6.4
3.-continued
(d) Suppose  is an integer where 1 < , x is an integer where 0 < x < ,
n is an integer where 0 < n <   x, and  = 2. Find each of the following:
o
ex=
o
e x : n =
ex =
e x : n =
9
Sections 6.1, 6.2, 6.3, 6.4
3.-continued
(e) Suppose  = 10 and  = 2. Find each of the following:
o
e4=
o
e 4 : 3 =
e4 =
e4 : 3 =
10
Sections 6.1, 6.2, 6.3, 6.4
4. Let T0 be a lifetime random variable with survival function
S0(t) = (1  t / ) for 0  t  
where  and  are a positive constants.
(a) Find a formula for each of E(T0), E(T02), and Var(T0).
11
Sections 6.1, 6.2, 6.3, 6.4
4.-continued
(b) Let Tx be the future lifetime random variable for an entity age x. Find
a formula for each of E(Tx), E(Tx2), and Var(Tx).
12
Sections 6.1, 6.2, 6.3, 6.4
4.-continued
(c) Suppose  = 2 and  = 10. Find each of the following:
3q2
=
2q3
=
q3 =
2 | q3
=
2 | 4 q3
=
4 | 2 q3
=
3q7
=
3p2
=
4p5
=
p8 =
4p8
=
13
Sections 6.1, 6.2, 6.3, 6.4
4.-continued
(d) Suppose  is an integer where 1 < , x is an integer where 0 < x < 
and n is an integer where 0 < n <   x. Find each of the following:
o
ex=
o
e x : n =
ex =
ex : n =
14
Sections 6.1, 6.2, 6.3, 6.4
4.-continued
(e) Suppose  = 10 and  = 1.5. Find each of the following:
o
e4=
o
e 4 : 3 =
e4 =
e 4 : 3 =
15