CHAPTER 4 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 4 Section 1 Problem 1
A parabolic satellite dish reflects signals to the dish’s focal point. An antenna designer analyzed
signals transmitted to a satellite dish and obtained the probability density function
( )
2
1
116
f x c x
=−
for
02x
, where X is the distance (in meters) from the centroid of the
dish surface to a reflection point at which a signal arrives. Determine the following:
(a) Value of c that makes
( )
fx
a valid probability density function
(b)
( )
0.4PX=
(c)
(0.1 0.4)PX =
SOLUTION
(a)
Reserve Problems Chapter 4 Section 1 Problem 2
The talk time (in hours) on a cell phone in a month is approximated by the probability density
function
( )
10
5
x
fx h
−
=
for
10 15x
,
1
h
for
15 20x
,
25
5
x
h
−
−
for
20 25x
.
Determine the following:
(a)
h=
(b)
( 17.5)PX=
(c)
( 22.0)PX=
(d) x such that
( ) 0.95P X x=
SOLUTION
02x
(a)
Let a continuous random variable X denote the time spent on a cell phone per month. Since
10 25x
, the probability density function should meet the condition:
( )
25
10 d1f x x=
.
Reserve Problems Chapter 4 Section 1 Problem 3
A test instrument needs to be calibrated periodically to prevent measurement errors. After some
time of use without calibration, it is known that the probability density function of the
measurement error is
( ) ( )
1.0 1 0.5f x x=−
for
0 2.0x
millimeters. Note that x is the
absolute value of the measurement error.
(a) If the measurement error within 0.5 millimeters is acceptable, what is the probability that the
error is not acceptable and the instrument needs calibration?
(b) What is the value of measurement error that must be exceeded with probability 0.3before the
instrument needs calibration?
(c) What is the probability that the measurement error is exactly 0.21millimeters before
calibration?
SOLUTION
(a)
(b)
(c)
( )
Reserve Problems Chapter 4 Section 1 Problem 4
If X is a continuous random variable, argue that
( )
1 2 1 2 1 2 1 2
( ) ( ) ( )P x X x P x X x P x X x P x X x = = =
SOLUTION
Reserve Problems Chapter 4 Section 2 Problem 1
The waiting line at a popular bakery shop can be quite long. Suppose that the waiting time in
minutes has probability density function
( )
0.1
0.1 x
f x e−
=
. Determine the following:
(a)
( )
Fx
(b) Probability that a customer waits more than 20 minutes:
(c) Probability that a customer waits less than 4 minutes:
SOLUTION
(a)
Let a continuous random variable X denote the waiting time in minutes. By definition, the
cumulative distribution function of a continuous random variable X is
Reserve Problems Chapter 4 Section 2 Problem 2
The coverage of a base station of a telecommunication company forms a disk with a radius of R
(kilometers). Let X be the distance of a cellphone from a base station. Assume that the location
of cellphones in use are randomly uniformly distributed within the disk. Determine the
following:
(a)
( )
Fx
(b)
( )
fx
(c)
( 0.4 )P x R=
Reserve Problems Chapter 4 Section 2 Problem 3
Determine the cumulative distribution function for
( )
1000
1
1000
x
fx
−
=
( 0)x
. Use the
cumulative distribution function to determine the probability that a component lasts more than
1000 hours before failure.
Reserve Problems Chapter 4 Section 2 Problem 4
( )
fx
Determine the cumulative distribution function for the random variable with the probability
density function
( )
0.500 0.125 , 0 < 4.f x x x= −
.
Reserve Problems Chapter 4 Section 3 Problem 1
A parabolic satellite dish reflects signals to the dish’s focal point. An antenna designer analyzed
signals transmitted to a satellite dish and obtained the probability density function
( )
2
1
116
f x c x
=−
for
03x
, where x is the distance (in meters) from the centroid of the
dish surface to a reflection point at which a signal arrives. Calculate the mean and variance.
( )
EX =
( )
VX=
SOLUTION
Let a continuous random variable X denote the distance (in meters) from the centroid of the dish
surface to a reflection point. Since
03x
, probability density function should meet the
condition:
( )
3
0d1f x x=
.
Reserve Problems Chapter 4 Section 3 Problem 2
The talk time (in hours) on a cell phone in a month is approximated by the probability density
function
( )
10
5
x
fx h
−
=
for
10 15x
,
1
h
for
15 20x
,
25
5
x
h
−
−
for
20 25x
. Determine
the following:
( )
EX =
( )
VX=
SOLUTION
Let a continuous random variable X denote the time spent on a cell phone per month. Since
10 25x
, probability density function should meet the condition:
( )
25
10 d1f x x=
.
The probability density function for X is:
10 ,
5
x
h
−
10 15x
Reserve Problems Chapter 4 Section 3 Problem 3
The waiting line at a popular bakery shop can be quite long. Suppose that the waiting time in
minutes has probability density function
( )
0.1
0.1 .
x
f x e−
=
Calculate the mean and variance for
the random variable:
( )
EX =
( )
VX=
SOLUTION
Let a continuous random variable X denote the waiting time in minutes. The probability density
function for X is
( )
0.1
0.1 ,0
x
f x e x
−
= +
.
Reserve Problems Chapter 4 Section 3 Problem 4
The coverage of a base station of a telecommunication company forms a disk with a radius of R
(kilometers). Let X be the distance of a cellphone from a base station. Assume that the location
of cellphones in use are randomly uniformly distributed within the disk. Calculate the mean and
variance for the random variable.
( )
EX =
( )
VX=
SOLUTION
Let a continuous variable X denote the distance of a cell phone from a base station (the range of
X is from zero to R). The probability density function for X is
( )
2
2x
fx R
=
.
Reserve Problems Chapter 4 Section 3 Problem 5
Determine the mean and variance of the random variable for
( )
( )
2
68 , 0 6
432
xx
f x x
−
=
.
SOLUTION
Reserve Problems Chapter 4 Section 3 Problem 6
Determine the mean and variance of the random variable with the probability density function
( ) ( )
1.6 1 0.8f x x=−
,
0 1.25x
.
SOLUTION
Reserve Problems Chapter 4 Section 4 Problem 1
Suppose the difference between actual and predicted weekly sales of a company follows a
uniform distribution between -$4000 and $4000. Determine the following:
(a) The probability that the actual sale is under predicted by at least $1400?
(b) The probability that the actual sale is within $1500 of the predicted sale?
SOLUTION
Let X denote the difference between the actual and predicted weekly sales of a company
( )
actual sale predicted salex=−
.
Reserve Problems Chapter 4 Section 4 Problem 2
In a paper manufacturing company, a machine is used to press wet fiber web into a continuous
roll of paper. This machine does not create a constant pressure on wet fiber web and final sheets
of papers have different thickness which is uniformly distributed between 0.004 and 0.015 inch.
Let X denote the thickness of the sheet of paper. Determine the following:
(a) Mean and variance for thickness of each paper sheet.
(b) Proportion of paper sheets which are less than 0.0095 inch thick.
(c) Thickness exceeded by 40 percent of the paper sheets.
SOLUTION
(a)
0.011
Reserve Problems Chapter 4 Section 4 Problem 3
A beacon transmits a signal every 10 minutes (such as 8:20, 8:30, etc.). The time at which a
receiver is tuned to detect the beacon is a continuous uniform distribution from 8:00 a.m. to 9:00
a.m. Consider the waiting time until the next signal from the beacon is received.
(a) Is it reasonable to model the waiting time as a continuous uniform distribution?
(b) What is the mean waiting time?
(c) What is the probability that the waiting time is less than 4 minutes?
SOLUTION
(a)
Reserve Problems Chapter 4 Section 5 Problem 1
A credit card company monitors cardholder transaction habits to detect any unusual activity.
Suppose that the dollar value of unusual activity for a customer in a month follows a normal
distribution with mean $250 and variance $391.
(a) What is the probability of $250 to $300 in unusual activity in a month?
(b) What is the probability of more than $300 in unusual activity in a month?
(c) Suppose that 10 customer accounts independently follow the same normal distribution. What
is the probability that at least one of these customers exceeds $300 in unusual activity in a
month?
SOLUTION
(a)
Let X denote the dollar value of an unusual activity for a customer per month. Then X is a normal
Reserve Problems Chapter 4 Section 5 Problem 2
A laptop company claims up to 9.1 hours of wireless web usage for its newest laptop battery life.
However, reviews on this laptop shows many complaints about low battery life. A survey on
battery life reported by customers shows that it follows a normal distribution with mean 8.5
hours and standard deviation 39 minutes.
(a) What is the probability that the battery life is at least 9.1 hours?
(b) What is the probability that the battery life is less than 7.9 hours?
(c) What is the time of use that is exceeded with probability 0.9?
SOLUTION
(a)
Reserve Problems Chapter 4 Section 5 Problem 3
Use Appendix Table III to determine the following probabilities for the standard normal variable
Z.
(a)
( 0.7 0.7)PZ− =
(b)
( 1.5 1.5)PZ− =
(c)
( 2.0 2.0)PZ− =
(d)
( 2.0)PZ=
(e)
(0 0.7)PZ =
SOLUTION
Reserve Problems Chapter 4 Section 5 Problem 4
Assume that X is normally distributed with a mean of 7 and a standard deviation of 4. Determine
the value for x that solves each of the following equations.
(a)
( ) 0.5P X x=
(b)
( ) 0.95P X x=
(c)
( 9) 0.2P x X =
(d)
(3 ) 0.95P X x =
(e)
( 7 ) 0.99P x X x− − =
SOLUTION
Reserve Problems Chapter 4 Section 5 Problem 5
The fill volume of an automated filling machine used for filling cans of carbonated beverage is
normally distributed. Suppose, that the mean of the filling operation can be adjusted easily, but
the standard deviation remains at 0.3 fluid ounce.
(a) At what value should the mean be set so that 99.9% of all cans exceed 12 fluid ounces?
(b) At what value should the mean be set so that 99.9% of all cans exceed 12 fluid ounces if the
standard deviation can be reduced to 0.07 fluid ounce?
SOLUTION
(a)
Reserve Problems Chapter 4 Section 5 Problem 6
The speed of a file transfer from a server on campus to a personal computer at a student’s home
on a weekday evening is normally distributed with a mean of 63 kilobits per second and a
standard deviation of four kilobits per second.
(a) What is the probability that the file will transfer at a speed of 70 kilobits per second or more?
(b) What is the probability that the file will transfer at a speed of less than 57 kilobits per
second?
(c) If the file is one megabyte, what is the average time (in seconds) it will take to transfer the
file? (Assume eight bits per byte)
SOLUTION
(a)
Reserve Problems Chapter 4 Section 5 Problem 7
Measurement error that is normally distributed with a mean of 0 and a standard deviation of 0.5
gram is added to the true weight of a sample. Then the measurement is rounded to the nearest
gram. Suppose that the true weight of a sample is 167.4 grams.
(a) What is the probability that the rounded result is 167 grams?
(b) What is the probability that the rounded result is 167 grams or more?
SOLUTION
(a)
Reserve Problems Chapter 4 Section 5 Problem 8
A study by Bechtel et al., 2009, described in the Archives of Environmental & Occupational
Health considered polycyclic aromatic hydrocarbons and immune system function in beef cattle.
Some cattle were near major oil- and gas-producing areas of western Canada. The mean monthly
exposure to PM1.0 (particulate matter that is
1m
in diameter) was approximately 7.2
3
/mg
with standard deviation 1.5. Assume that the monthly exposure is normally distributed.
(a) What is the probability of a monthly exposure greater than 9
3
/mg
?
(b) What is the probability of a monthly exposure between 3 and 5
3
/mg
?
(c) What is the monthly exposure level that is exceeded with probability 0.05?
(d) What value of mean monthly exposure is needed so that the probability of a monthly
exposure more than 9
3
/mg
is 0.01?
SOLUTION
(a)
(c)
( )
(d)
Reserve Problems Chapter 4 Section 5 Problem 9
An article in International Journal of Electrical Power & Energy Systems [“Stochastic Optimal
Load Flow Using a Combined Quasi-Newton and Conjugate Gradient Technique” (1989, Vol.
11(2), pp. 85-93)] considered the problem of optimal power flow in electric power systems and
included the effects of uncertain variables in the problem formulation. The method treats the
system power demand as a normal random variable with 0 mean and unit variance.
(a) What is the power demand value exceeded with 80% probability?
(b) What is the probability that the power demand is positive?
(c) What is the probability that the power demand is greater than –1 and less than 1?
SOLUTION
(a)
Reserve Problems Chapter 4 Section 6 Problem 1
City planners estimate that they can attract 20% of commuters who currently use bus service to a
new light rail to the downtown area. Suppose that 71000 independent commuters use bus service
daily.
(a) Approximate the probability that more than 10300 commuters will use the light rail daily.
(b) Approximate the number of daily commuters who will use the light rail which will be
exceeded with probability only 0.05.
The number of daily commuters who will use the light rail which will be exceeded with
probability only 0.05 is _______.
SOLUTION
(a)
Reserve Problems Chapter 4 Section 6 Problem 2
Disaster data can be used to inform policy decisions to help reduce disaster risks and build
resilience. Geological investigations and historical record of earthquakes in New Zealand show
that the average number of earthquakes with magnitude 4-4.9 is 445 per year. Assume a Poisson
distribution.
(a) Approximate the probability that at least 505 earthquakes occur in a year.
(b) Approximate the number of earthquakes exceeded with probability 0.90.
SOLUTION
(a)
Reserve Problems Chapter 4 Section 6 Problem 3
Suppose that X has a Poisson distribution with a mean of 70. Approximate the following
probabilities.
a)
( 78)PX
with and without using continuity correction.
b)
( 70)PX
with and without using continuity correction.
c)
(66 74)PX
with and without using continuity correction.
SOLUTION
Reserve Problems Chapter 4 Section 6 Problem 4
Among homeowners in a metropolitan area, 75% recycle plastic bottles each week. A waste
management company services 1700 homeowners (assumed independent). Approximate the
following probabilities.
a) At least 1300.00 recycle plastic bottles in a week.
b) Between 1225.00 and 1325.00 recycle plastic bottles in a week.
SOLUTION
Reserve Problems Chapter 4 Section 7 Problem 1
An article in the European Physical Journal B [“Evidence for the exponential distribution of
income in the USA” (2001, Vol. 20, pp. 585-589)] shows that individual income in the US can
be approximated with an exponential distribution. Suppose that the mean of individual income in
a year is $22000.
(a) Write the probability distribution function for individual income.
(b) What percentage of individuals in the US earn less than $21000?
(c) What percentage of individuals in the US earn more than $50000?
(d) What is the income exceeded by 2% of individuals?
SOLUTION