Reserve Supplemental Exercises Chapter 5 Problem 6
To evaluate the technical support from a computer manufacturer, the number of rings before a
call is answered by a service representative is tracked. Historically, 70% of the calls are
answered in two rings or less, 25% are answered in three or four rings, and the remaining calls
require five rings or more. Suppose that you call this manufacturer 10 times and assume that the
calls are independent. Let X, Y, and Z denote the number of calls answered in two rings or less, in
three or four rings, and in five rings or more, respectively.
(a) What is the probability that 7 calls are answered in two rings or less, 2 cells are answered in
three or four rings, and one call requires five rings or more?
(b) Let W denote the number of calls answered in four rings or less. What is the probability that
all 10 calls are answered in four rings or less?
(c) What is the expected number of calls answered in four rings or less?
(d) What is the conditional distribution of the number of calls requiring five rings or more given
that eight calls are answered in two rings or less?
(e) What is the conditional expected number of calls requiring five rings or more given that eight
calls are answered in two rings or less?
(f) Are the number of calls answered in two rings or less and the number of calls requiring five
rings or more independent random variables?
SOLUTION
(a)
(e)
Reserve Supplemental Exercises Chapter 5 Problem 7
The length and width of panels used for interior doors (in inches) are denoted as X and Y,
respectively. Suppose that X and Y are independent, continuous uniform random variables for
17.75 18.25x
and
4.75 5.25y
, respectively.
(a) By integrating the joint probability density function over the appropriate region, determine
the probability that the area of a panel exceeds 93 square inches.
(b) What is the probability that the perimeter of a panel exceeds 46 inches?
SOLUTION
(a)
(b)
Reserve Supplemental Exercises Chapter 5 Problem 8
If determine
( )
EX
,
( )
EY
,
( )
VX
,
( )
VY
, and
by reorganizing the parameters in the joint
probability density function.
( )
EX
=
( )
EY
=
( )
VX
=
( )
VY
=
=
SOLUTION
Reserve Supplemental Exercises Chapter 5 Problem 9
Use moment generating functions to determine the normalized power
( )
4 1/4
[]EX
from a cycling
power meter when X has a normal distribution with mean 210 and standard deviation 20 Watts.
SOLUTION
The moment generating function for a normal distribution is
Reserve Supplemental Exercises Chapter 5 Problem 10
Show that if
1
X
,
2
X
, …,
p
X
are independent, continuous random variables,
( )
( ) ( )
( )
1 1 2 2 1 1 2 2
, ,…, …
p p p p
P X A X A X A P X A P X A P X A =
for any regions
1
A
,
2
A
,
…,
p
A
in the range of
1
X
,
2
X
, …,
p
X
, respectively. To do so, complete the following
derivation by selecting the correct answers.
By the ______,
( ) ( )
1 2 1 2
1 1 2 2 … 1 2 1 2
, ,…, … , ,…, d d …d
pp
p p A A A X X X p p
P X A X A X A f x x x x x x =
From the _______,
( )
( ) ( )
( )
1 2 1 2
… 1 2 1 2
, ,…, …
pp
X X X p X X X p
f x x x f x f x f x=
Therefore,
( )
( ) ( )
( )
1 2 1 2 1 2
12
… 1 2 1 2 1 1 2 2
… , ,..., d d ...d d d ... d
p p p
p
A A A X X X p p X X X p p
A A A
f x x x x x x f x x f x x f x x
=
By the _______,
( ) ( )
( )
( ) ( )
( )
12
12
1 1 2 2 1 1 2 2
d d … d …
p
p
X X X p p p p
A A A
f x x f x x f x x P X A P X A P X A
=
SOLUTION
By the definition,
Reserve Supplemental Exercises Chapter 5 Problem 11
Show that if
1
X
,
2
X
, …,
p
X
are independent random variables and
1 1 2 2 … pp
Y c X c X c X= + + +
, then
( ) ( ) ( )
( )
2 2 2
1 1 2 2 … pp
V Y c V X c V X c V X= + + +
.
SOLUTION
By the definition,
Reserve Supplemental Exercises Chapter 5 Problem 12
Suppose that the joint probability function of the continuous random variables X and Y is
constant on the rectangle
0xa
,
0yb
. Show that X and Y are independent.
Express your answers in terms of a and b. Determine the constant c such that
( )
,
XY
f x y c=
for
0xa
,
0yb
.
SOLUTION
Reserve Supplemental Exercises Chapter 5 Problem 13
Suppose that the range of the continuous variables X and Y is
0xa
and
0yb
. Also
suppose that the joint probability density function
( ) ( ) ( )
,
XY
f x y g x h y=
where
( )
gx
is a
function only of x, and
( )
hy
is a function only of y. Show that X and Y are independent.
SOLUTION
The marginal probability density function of X is
Reserve Supplemental Exercises Chapter 5 Problem 14
This exercise extends the hypergeometric distribution to multiple variables. Consider a
population with N items of k different types. Assume that there are
1
N
items of type 1,
2
N
items of type 2, …,
k
N
items of type k so that
12
… k
N N N N+ + + =
. Suppose that a random
sample of size n is selected, without replacement, from the population. Let
1
X
,
2
X
, …,
k
X
denote the number of items of each type in the sample so that
12
… n
X X X n+ + + =
. Show that
for feasible values of n,
1
x
,
2
x
, …,
k
x
,
1
N
,
2
N
, …,
k
N
, the probability is
( )
12
12
1 1 2 2
…
, ,…,
k
k
kk
N
NN
x
xx
P X x X x X x N
n
= = = =
.
SOLUTION
Reserve Supplemental Exercises Chapter 5 Problem 15
Use the properties of moment-generating functions to show that a sum of P independent normal
random variables with means
i
and variances
2
i
for
1, 2,…,ip=
has a normal distribution.
Find mean and variance for this distribution.
The moment generating function of a normal random variable with mean
and standard
deviation
is ____.
Mean
of the sum of p independent normal random variables is equal to _____.
Variance
2
of the sum of p independent normal random variables is equal to _____.
SOLUTION
( )
12
12
1 1 2 2
…
, ,…,
k
k
kk
N
Reserve Supplemental Exercises Chapter 5 Problem 16
Show that by expanding
tX
e
in a power series and taking expectations term by term we may
write the moment-generating function as
( )
( )
2
12
1 … ...
2! !
r
tX s s s
Xr
tt
M t E e t r
= = + + + + +
. Thus,
the coefficient of
!
r
t
r
in this expansion is
s
r
, the rth origin moment. Use the power series
expansion for
( )
X
Mt
for a gamma random variable to determine
1
s
and
2
s
.
SOLUTION
