CHAPTER 5 RESERVE PROBLEMS
The following problems have been reserved for your use in assignments and testing and do not
appear in student versions of the text.
Reserve Problems Chapter 5 Section 1 Problem 1
Let the random variable X denote the time until a computer server connects to your machine (in
milliseconds), and let Y denote the time until the server authorizes you as a valid user (in
milliseconds). Each of these random variables measures the wait from a common starting time
and
XY
. Assume that the joint probability density function for X and Y is
( )
6
6 10 exp 0.001 0.002
XY
f x y
−
= − −
for
xy
. Determine the probability that
2600XY+
.
SOLUTION
The probability that
0XY
and
2600XY+
is determined as
6 0.001 0.002
2600
( 2600) 6 10 xy
xy
P X Y e dxdy
− − −
+
+ = ∬
Reserve Problems Chapter 5 Section 1 Problem 2
In the article “Joint distribution model for prediction of hurricane wind speed and size” (2012,
Structural Safety, 35, 40–51) the authors characterized the joint distribution of hurricane
maximum wind speed and size. According to the study, 99% of the hurricanes in the past 50
years were approximately up to 30 m/s maximum wind speed (V) and 50 km size (R). Suppose
combinations of wind speed and size are uniformly distributed within the ellipse
2 2 2 2 2
50 30 (50 30)vr+ =
, and
0 30v
and
0 50r
. Determine the following:
(a)
( )
10, 35P V R =
(b)
( )
10PV=
(c)
( )
ER=
SOLUTION
(a)
(b)
(c)
Reserve Problems Chapter 5 Section 1 Problem 3
Two methods of measuring surface smoothness are used to evaluate a paper product. The
measurements are recorded as deviations from the nominal surface smoothness in coded units.
The joint probability distribution of the two measurements is a uniform distribution over the
region
04x
,
0y
, and
11x y x− +
. That is,
( )
,
XY
f x y c=
for X and Y in the region.
Determine the value for c such that
( )
,
XY
f x y
is a joint probability density function.
c=
(a)
( 0.5, 0.6)P X Y =
(b)
( 0.5)PX=
(c)
( )
EX =
(d)
( )
EY =
(e) For
01x
marginal probability distribution of X is ________.
SOLUTION
The value of c.
( )
1 1 4 1 1 4
0 0 1 1 0 1
3
d d d d 1 d 2 1d 6 7.5 1
xx
x
c y x c y x c x x c x c c c
++
−
+ = + + = + = =
.
Reserve Problems Chapter 5 Section 1 Problem 4
Let c be the constant, that makes
( )
,,
XYZ
f x y z c=
a joint probability density function over the
region
0, 0, 0 and 1x y z x y z + +
. Determine the following:
c=
(a)
( 0.1, 0.1, 0.1)P X Y Z =
(b)
( 0.5, 0.3)P X Y =
(c)
( 0.2)PX=
(d)
( )
EX =
(e) Marginal distribution of X
(f) Joint distribution of X and Y
(g) Conditional probability distribution of X given that
0.5Y=
SOLUTION
Determine c such that
( )
,,f x y z c=
is a joint density probability over the region
0, 0, 0x y z
with
1x y z+ +
.
Reserve Problems Chapter 5 Section 2 Problem 1
Consider joint probability distribution given below.
x
y
( )
,
XY
f x y
1.0
1
1/32
1.5
2
11/32
1.5
3
1/4
2.5
4
1/4
3.0
5
1/8
Determine the following:
(a) Conditional probability distribution of Y given that
1.5X=
y
( )
|1.5Y
fy
1
2
3
4
5
(b) Conditional probability distribution of Y given that
2X=
x
( )
|2X
fy
1.0
1.5
2.5
3
(c)
( | 1.5)E Y X ==
(d) Are X and Y independent?
SOLUTION
Reserve Problems Chapter 5 Section 2 Problem 2
Consider the following joint probability distribution:
x
y
( )
,
XY
f x y
–1.0
–2
1/8
–0.4
–1
1/4
0.4
1
1/16
1.0
2
9/16
Determine the following:
(a) Conditional probability distribution of Y given that
1X=
( )
|1Y
fy=
for
y=
(b) Conditional probability distribution of X given that
1Y=
( )
|1X
fx=
for x =
(c)
( | 1)E X Y ==
(d) Are X and Y independent?
SOLUTION
Reserve Problems Chapter 5 Section 2 Problem 3
In the transmission of digital information, the probability that a bit has high, moderate, and low
distortion is 0.01, 0.07, and 0.92, respectively. Suppose that three bits are transmitted and that
the amount of distortion of each bit is assumed to be independent. Let X and Y denote the number
of bits with high and moderate distortion out of the three, respectively. Determine the following:
(a)
( )
|1Y
fy
y
( )
|1Y
fy
0
1
2
3
(b)
( | 1)E Y X ==
(c) Are X and Y independent?
SOLUTION
(a) The joint probability mass function of the discrete random variables X and Y, denoted as
( )
,
XY
f x y
, satisfies
|1 00
YY
Reserve Problems Chapter 5 Section 2 Problem 4
A small-business Web site contains 100 pages and 52%, 32%, and 16% of the pages contain low,
moderate, and high graphic content, respectively. A sample of four pages is selected without
replacement, and X and Y denote the number of pages with moderate and high graphics output in
the sample. Determine the following:
(a)
( )
|3Y
fy
y
( )
|3Y
fy
0
1
2
3
4
(b)
( | 3)E Y X ==
(c)
( | 3)V Y X ==
(d) Are X and Y independent?
SOLUTION
(a)
The range of (X, Y) is
0X
,
0Y
and
4XY+
. Here X is the number of pages with
moderate graphic content and Y is the number of pages with high graphic output among a sample
0
0.7648
1
0.2353
2
0
3
0
4
0
Reserve Problems Chapter 5 Section 2 Problem 5
A manufacturing company employs two devices to inspect output for quality control purposes.
The first device is able to accurately detect 99.5% of the defective items it receives, whereas the
second is able to do so in 99.8% of the cases. Assume that four defective items are produced and
sent out for inspection. Let X and Y denote the number of items that will be identified as
defective by inspecting devices 1 and 2, respectively. Assume that the devices are independent.
(a)
( )
|2Y
fy
y
fY|2(y)
0
1
2
3
4
(b)
( | 2)E Y X ==
(c)
( | 2)V Y X ==
(d) Are X and Y independent?
SOLUTION
y
Reserve Problems Chapter 5 Section 2 Problem 6
In the article “Joint distribution model for prediction of hurricane wind speed and size” (2012,
Structural Safety, 35, 40–51) the authors characterized the joint distribution of hurricane
maximum wind speed and size. According to the study, 99% of the hurricanes in the past 50
years were approximately up to 30 m/s maximum wind speed (V) and 50 km size (R) Suppose
combinations of wind speed and size are uniformly distributed within the ellipse
2 2 2 2 2
50 30 (50 30)vr+ =
, and
0 30v
and
0 50r
. Determine the following:
(a)
( 20 | 20 30)P R V =
(b)
( )
15PV=
(c)
( )
15, 25P V R =
SOLUTION
(a)
The distribution is uniform, so its joint probability density function is constant and equal to the
inverse value of the one fourth of the area of the ellipse:
Reserve Problems Chapter 5 Section 2 Problem 7
Two methods of measuring surface smoothness are used to evaluate a paper product. The
measurements are recorded as deviations from the nominal surface smoothness in coded units.
The joint probability distribution of the two measurements is a uniform distribution over the
region
04x
,
0y
, and
11x y x− +
. That is,
( )
,
XY
f x y c=
for x and y in the region.
Determine the value for c such that
( )
,
XY
f x y
is a joint probability density function.
c=
a) Conditional probability distribution of Y given
1X=
b)
( | 1.5)E Y X ==
c)
( )
0.8 | 1.5P Y X = =
SOLUTION
The value of c.
Reserve Problems Chapter 5 Section 4 Problem 1
In the article “Joint distribution model for prediction of hurricane wind speed and size” (2012,
Structural Safety, 35, 40–51) the authors characterized the joint distribution of hurricane
maximum wind speed and size. According to the study, 99% of the hurricanes in the past 50
years were approximately up to 30 m/s maximum wind speed (V) and 50 km size (R). Suppose
combinations of wind speed and size are uniformly distributed within the ellipse
2 2 2 2 2
50 30 (50 30)vr+ =
, and
0 30v
and
0 50r
. Calculate the covariance and
correlation between maximum wind speed and hurricane size.
( )
cov ,RV =
RV
=
SOLUTION
The distribution is uniform, so its joint probability density function is constant and equal to the
inverse value of the one fourth of the area of the ellipse:
( )
4
,30 50
VR
f v r
=
Reserve Problems Chapter 5 Section 4 Problem 2
Suppose that the random variables X, Y, and Z have the joint probability density function
( )
,,
XYZ
f x y z c=
over the cylinder
22
9xy+
and
03z
. Calculate the covariance and
correlation between all pairs of the random variables.
( )
cov ,XZ =
XZ
=
( )
cov ,YZ =
YZ
=
( )
cov ,XY =
XY
=
SOLUTION
Reserve Problems Chapter 5 Section 4 Problem 3
Suppose that X and Y are independent continuous random variables. Show that
0
XY
=
.
SOLUTION
Reserve Problems Chapter 5 Section 5 Problem 1
Suppose that X and Y are the number of conforming and nonconforming parts, respectively, in a
sample of size 20, selected randomly, without replacement from a batch of 100 which contain n
conforming parts. Is the joint distribution of X and Y multinomial?
SOLUTION
Reserve Problems Chapter 5 Section 5 Problem 2
In an online gaming competition, you might score in the top, middle, or low category in each
match with probabilities 0.3, 0.6, and 0.1, respectively. Among 10 independent games, let X, Y,
and Z denote the number scored in the top, middle, and low category, respectively. Determine
the following:
(a)
( 3)PX=
(b)
( 10)P X Y+ =
(c)
( )
3| 5P X Z = =
(d)
( )
5 | 3P X Y Z+ = =
(e)
( 3 | 2)P X Z =
SOLUTION
(a) Because X has a binomial distribution with
0.3p=
Reserve Problems Chapter 5 Section 5 Problem 3
Test results from an electronic circuit board indicate that 50% of board failures are caused by
assembly defects, 30% by electrical components, and 20% by mechanical defects. Suppose that
10 boards fail independently. Let the random variables
, ,andX Y Z
denote the number of
assembly, electrical, and mechanical defects among the 10 boards. Calculate the following:
(a)
( )
5, 4, 1P X Y Z= = = =
(b)
( )
8PX==
(c)
( 8 | 1)P X Y= = =
(d)
( 8 | 1)P X Y = =
(e)
( 7, 1| 2)P X Y Z= = = =
SOLUTION
(a)
Board failures caused by assembly defects with probability
10.5p=
,
Reserve Problems Chapter 5 Section 5 Problem 4
Four electronic ovens that were dropped during shipment are inspected and classified as
containing either a major, a minor, or no defect. In the past, 60% of dropped ovens had a major
defect, 30% had a minor defect, and 10% had no defect. Assume that the defects on the four
ovens occur independently.
(a) What is the probability that, of the four dropped ovens, two have a major defect and two have
a minor defect?
(b) What is the probability that no oven has a defect?
(c) Let X and Y denote the numbers of ovens with major and minor defects, respectively. Joint
probability mass function of the number of ovens with a major defect and the number with a
minor defect:
(d) Expected number of ovens with a major defect:
(e) Expected number of ovens with a minor defect:
(f) Conditional probability that two ovens have major defects given that two ovens have minor
defects:
(g) Conditional probability that three ovens have major defects given that two ovens have minor
defects:
(h) Conditional probability distribution of the number of ovens with major defects (X) given that
two i. ovens have minor defects (Y):
(i) Conditional mean of the number of ovens with major defects (X) given that two ovens have
minor defects (Y):
SOLUTION
(a)
Let X, Y, and Z denote the number of ovens in the sample of four with major, minor, and no
Reserve Problems Chapter 5 Section 5 Problem 5
Suppose that X has a standard normal distribution. Let the conditional distribution of Y given
Xx=
be normally distributed with mean
( | ) 2E Y X x=
and variance
( | ) 2V Y x =
. Determine
the following:
(a) Are X and Y independent?
(b)
( )
3| 3P Y X = =
(c)
( | 1)E Y X ==
(d)
( )
,
XY
f x y =
(e) Identify the mean and variance of Y and the correlation between X and Y.
SOLUTION
(a)