Section 5.3 Class Exercises
Math 400 – Actuarial Models
NAME:_____________________________
1. Let T0 be a lifetime random variable having a uniform distribution on the
interval from 0 to .
(a) Find a formula for each of the following:
S0(t) =
f0(t) =
0(t) = t =
E(T0) =
1
Section 5.3 Class Exercises
1.-continued
(b) For a given value x, let Tx be the remaining lifetime random variable
given that T0 = x. Find a formula for each of the following:
t
px =
t
qx =
fx (t) =
x (t) = x + t =
E(Tx) =
2
Section 5.3 Class Exercises
2. Let T0 be a lifetime random variable with survival function
S0(t) = 1  (t / )k for 0  t  
where k is a positive constant.
(a) Find a formula for each of the following:
F0(t) =
f0(t) =
0(t) = t =
E(T0) =
3
Section 5.3 Class Exercises
2.-continued
(b) For a given value x, let Tx be the remaining lifetime random variable
given that T0 = x. Find a formula for each of the following:
t
px =
t
qx =
fx (t) =
x (t) = x + t =
E(Tx) =
4
Section 5.3 Class Exercises
3. Let T0 be a lifetime random variable with survival function
S0(t) = (1  t / )k for 0  t  
where k is a positive constant.
(a) Find a formula for each of the following:
F0(t) =
f0(t) =
0(t) = t =
E(T0) =
5
Section 5.3 Class Exercises
3.-continued
(b) For a given value x, let Tx be the remaining lifetime random variable
given that T0 = x. Find a formula for each of the following:
t
px =
t
qx =
fx (t) =
x (t) = x + t =
E(Tx) =
6
Section 5.3 Class Exercises
4. Let T0 be a lifetime random variable having an exponential distribution
with mean .
(a) Find a formula for each of the following:
F0(t) =
S0(t) =
f0(t) =
0(t) = t =
E(T0) =
7
Section 5.3 Class Exercises
4.-continued
(b) For a given value x, let Tx be the remaining lifetime random variable
given that T0 = x. Find a formula for each of the following:
t
px =
t
qx =
fx (t) =
x (t) = x + t =
E(Tx) =
8
Section 5.3 Class Exercises
5. Let T0 be a lifetime random variable with survival function
S0(t) = 1 / (1 + t)k for 0  t
where  > 0 and k > 1 are constants.
(a) Find a formula for each of the following:
F0(t) =
f0(t) =
0(t) = t =
E(T0) =
9
Section 5.3 Class Exercises
5.-continued
(b) For a given value x, let Tx be the remaining lifetime random variable
given that T0 = x. Find a formula for each of the following:
t
px =
t
qx =
fx (t) =
x (t) = x + t =
E(Tx) =
10
Section 5.3 Class Exercises
6. Let T0 be a lifetime random variable having a uniform distribution on the
interval from 0 to  (i.e., the same distribution as in #1). Suppose  = 80.
(a) Find the p.d.f. for K0 and the p.d.f. for K0* .
(b) Find the mean and variance for each of K0 and K0* .
(c) Find the p.d.f. for K30 and the p.d.f. for K30* .
(d) Find the mean and variance for each of K30 and K30* .
11
Section 5.3 Class Exercises
6.-continued
(e) Let x be an integer such that 0 < x < 80. Find the p.d.f. for Kx, the
p.d.f. for Kx* , the mean and variance for Kx, and the mean and variance for
Kx* .
(f) Let  be an integer greater than 1, and let x be an integer such that
0 < x < . Find the p.d.f. for Kx , the p.d.f. for Kx* , the mean and variance
for Kx , and the mean and variance for Kx* .
12
Section 5.3 Class Exercises
7. Let T0 be a lifetime random variable with survival function
S0(t) = 1  (t / ) for 0  t  
where  is an integer greater than 1, and  is a positive constant (i.e., part of
the family of distributions in #2).
(a) Suppose  = 6 and  = 2. In the table below, complete the column
which defines the p.d.f. for K0 and the column which defines the p.d.f. for
K0* . Then, find E[K0] and E[K0*].
Possible
Possible
Values
Values
for K0
for K0*
Probability
____________________________________________________
0
1
13
Section 5.3 Class Exercises
7.-continued
(b) Suppose  = 6 and  = 2. In the table below, complete the column
which defines the p.d.f. for K2 and the column which defines the p.d.f. for
K2* . Then, find E[K2] and E[K2*].
Possible
Possible
Values
Values
for K2
for K2*
Probability
____________________________________________________
0
1
14
Section 5.3 Class Exercises
7.-continued
(c) Suppose  = 6,  = 2, and x is an integer such that 0 < x < 6. In the
table below, complete the column which defines the p.d.f. for Kx and the
column which defines the p.d.f. for Kx* . Then, find E[Kx] and E[Kx*].
Possible
Possible
Values
Values
for Kx
for Kx*
Probability
____________________________________________________
0
1
.
.
.
k1
k
.
.
.
E[Kx] =
6 x
 (k 1)
k 1
6 x
k
k 1
S 0 ( x  k  1)  S 0 ( x  k )
=
S 0 ( x)
S 0 ( x  k  1)  S 0 ( x  k )

S 0 ( x)
6 x

k 1
S0 ( x  k  1)  S0 ( x  k )
=
S 0 ( x)
15
Section 5.3 Class Exercises
6 x
 k(
k 1
k 1
px k px )  1 =
 ( j  1)
j 1
k
k 1
6 x 1
6 x 1
j
px 
k
k 1
6 x
6 x
p 
k 1 x
k
k 1
6 x 1
k
px =
j
j 1
px +
5 x
1  [( x  j ) / 6]
=
1  ( x / 6) 2
j 1

2
px  1 =

j 1
6 (5  x)   ( x  j ) 2
2
5 x
k
j 1
2
6 x
2
E[Kx*] =
16
k
k 2
p 
k 1 x
6 x 1
6 x 1
j
6 x 1
6 x
j
px 
k
k 1
k
k 1
k
px =
6 x 1
k
px =

j 1
j
px =
Section 5.3 Class Exercises
7.-continued
(d) Suppose  is an integer greater than 1,  is a positive constant, and x
is an integer such that 0 < x < . In the table below, complete the column
which defines the p.d.f. for Kx and the column which defines the p.d.f. for
Kx* . Then, using work already done in part (c), find E[Kx] and E[Kx*].
Possible
Possible
Values
Values
for Kx
for Kx*
Probability
____________________________________________________
0
1
.
.
.
k1
k
.
.
.
E[Kx] =
E[Kx*] =
17
Section 5.3 Class Exercises
8. Let T0 be a lifetime random variable with survival function
S0(t) = (1  t / ) for 0  t  
where  is an integer greater than 1, and  is a positive constant (i.e., part of
the family of distributions in #3).
(a) Suppose  = 6 and  = 2. In the table below, complete the column
which defines the p.d.f. for K0 and the column which defines the p.d.f. for
K0* . Then, find E[K0] and E[K0*].
Possible
Possible
Values
Values
for K0
for K0*
Probability
____________________________________________________
0
1
18
Section 5.3 Class Exercises
8.-continued
(b) Suppose  = 6 and  = 2. In the table below, complete the column
which defines the p.d.f. for K2 and the column which defines the p.d.f. for
K2* . Then, find E[K2] and E[K2*].
Possible
Possible
Values
Values
for K2
for K2*
Probability
____________________________________________________
0
1
19
Section 5.3 Class Exercises
8.-continued
(c) Suppose  = 6,  = 2, and x is an integer such that 0 < x < 6. In the
table below, complete the column which defines the p.d.f. for Kx and the
column which defines the p.d.f. for Kx* . Then, find E[Kx] and E[Kx*].
Possible
Possible
Values
Values
for Kx
for Kx*
Probability
____________________________________________________
0
1
.
.
.
k1
k
.
.
.
(To find the following expected values, use the work done in #7(c)&(d).)
E[Kx] =
E[Kx*] =
20
Section 5.3 Class Exercises
8.-continued
(d) Suppose  is an integer greater than 1,  is a positive constant, and x
is an integer such that 0 < x < . In the table below, complete the column
which defines the p.d.f. for Kx and the column which defines the p.d.f. for
Kx* . Then, find E[Kx] and E[Kx*].
Possible
Possible
Values
Values
for Kx
for Kx*
Probability
____________________________________________________
0
1
.
.
.
k1
k
.
.
.
(To find the following expected values, use the work done in #7(c)&(d).)
E[Kx] =
E[Kx*] =
21
Section 5.3 Class Exercises
9. Let T0 be a lifetime random variable having an exponential distribution
with mean  (i.e., the family of distributions in #4).
(a) Suppose k is a positive integer. Find a formula for P[K0 = k  1] =
P[K0* = k], and then find a formula for each of E[K0] and E[K0*].
(b) Suppose 0 < x, and k is a positive integer. Find a formula for
P[Kx = k  1] = P[Kx* = k], and then find a formula for each of E[Kx] and
E[Kx*].
22
Section 5.3 Class Exercises
10. Let T0 be a lifetime random variable with survival function
S0(t) = 1 / (1 + t) for 0  t
where  > 0 and  > 1 are constants (i.e., part of the family of distributions
in #5).
(a) Suppose  = 1,  = 2, and k is a positive integer. Find a formula for
P[K0 = k  1] = P[K0* = k], and then find each of E[K0] and E[K0*].

1 2
(Hint: To find the expected values, you will need to use  2 
).
6
n 1 n
23
Section 5.3 Class Exercises
10.-continued
(b) Suppose  = 1,  = 2, and k is a positive integer. Find a formula for
P[K3 = k  1] = P[K3* = k]; then find each of E[K3] and E[K3*].

1 2
(Hint: To find the expected values, you will need to use  2 
).
6
n 1 n
24