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(d)
Reserve Problems Chapter 4 Section 7 Problem 2
Web crawlers need to estimate the frequency of changes to Web sites to maintain a current index
for Web searches. Assume that the changes to a Web site follow a Poisson process with a mean
of 6 days. Let a random variable X denote the time (in days) until the next change.
a) What is the probability that the next change occurs in less than 4.5 days?
b) What is the probability that the time until the next change is greater 9.5 days?
c) What is the time of the next change that is exceeded with probability 90%?
d) What is the probability that the next change occurs in less than 12.5 days, given that it has not
yet occurred after 3.0 days?
SOLUTION
a)
Reserve Problems Chapter 4 Section 7 Problem 3
Requests for service in a queuing model follow a Poisson distribution with a mean of 4 per unit
time.
a) What is the probability that the time until the first request is less than 4 time units?
b) What is the probability that the time between the second and third requests is greater than 7.5
time units?
c) Determine the mean rate of requests such that the probability is 0.9 that there are no requests
in 0.5 time units?
d) If the service times are independent and exponentially distributed with a mean of 0.4 time
units, what can you conclude about the long-term response of this system to requests?
SOLUTION
(a)
Reserve Problems Chapter 4 Section 8 Problem 1
Human behavior can sometimes be modeled as a Poisson process. Suppose when you send a
tweet in Twitter, the retweets follow a Poisson process, and the mean time between retweets is 7
minutes.
(a) What is the probability that a retweet occurs less than 8 minutes after you send your tweet?
(b) How many retweets would you expect after 22 hours?
(c) What is the probability of more than four retweets in an hour after your tweet?
SOLUTION
(a)
(b)
Reserve Problems Chapter 4 Section 8 Problem 2
We model the gene mutation of a virus as a Poisson process. Suppose a virus mutates once every
3 months on average.
(a) What is mean time until the fifth mutation?
(b) What is the probability that more than five mutations occur in a year?
SOLUTION
(a)
(b)
Reserve Problems Chapter 4 Section 8 Problem 3
Show that the gamma density function
( )
,,f x r
integrates to 1.
SOLUTION
Reserve Problems Chapter 4 Section 9 Problem 1
An article in Journal of Wind Engineering and Industrial Aerodynamics [“Modern estimation of
the parameters of the Weibull wind speed distribution for wind energy analysis” 85(1), pages
75–84)] used the Weibull distribution to model the wind speed distribution. Suppose that
=
3
and
=
6 are used to model the wind speed (m/s) distribution.
Determine the following:
(a) Probability that the wind speed is between 4 m/s and 8 m/s.
(b) Mean and standard deviation of the wind speed.
SOLUTION
(a)
Let X denote the wind speed in m/s. Then, X is a Weibull random variable with
=
3 and
=
6.
Reserve Problems Chapter 4 Section 9 Problem 2
The time to failure (in days) of a starter motor in a vehicle is modeled as a Weibull distribution
with δ = 4000, β = 2.4.
(a) What is the probability that a starter motor fails within 4years?
(b) What is the probability a motor fails in less than 4 years given that is has lasted more than
two years?
(c) Compare your answer in the previous part to the probability that a motor fails in less than one
year and comment on whether the lack of memory property applies to this distribution.
The lack of memory property ________ to this distribution.
(d) Suppose that the time to failure (in days) of a new starter motor follows a Weibull
distribution with δ = 5000, β = 2.9 and that lifetime is defined as the time exceeded by 99% of
motors. Which motor has a longer lifetime?
SOLUTION
(a)
Reserve Problems Chapter 4 Section 9 Problem 3
Suppose that the lifetime of a component (in hours), X, is modeled with a Weibull distribution
with
0.5
=
and
3400
=
. Determine the following in parts (a) and (b).
a)
( 3500)PX=
b)
( 6000 3000)P X X =
c) Suppose that X has an exponential distribution with mean equal to 3400. Determine the
following probability.
Reserve Problems Chapter 4 Section 9 Problem 4
Suppose that X has a Weibull distribution with
2
=
and
2500
=
. Determine the following.
a)
( 5000)PX=
b) For an exponential random variable with the same mean as the Weibull distribution
( 5000)PX=
c) Comment on the probability that the lifetime exceeds 5000 hours under the Weibull and
exponential distributions.
SOLUTION
Reserve Problems Chapter 4 Section 10 Problem 1
An article in Proceedings of the 2007 International Symposium on Wikis [“Cooperation and
quality in Wikipedia” 2007, Pages (157 –164)] used the lognormal distribution to model the
number of edits per article during a period in Wikipedia. Most articles only have a small number
of edits, while a small number of articles have a disproportionately large number of edits. Over a
period of 240 weeks, suppose that the number of edits of the articles has approximately a
lognormal distribution with
2
=
and
1.5
=
.
(a) What is the probability that the number of edits exceeds 35?
(b) Determine the mean and standard deviation of the number of edits.
(c) What number of edits is exceeded by 80% of the articles?
SOLUTION
Reserve Problems Chapter 4 Section 10 Problem 2
The number of employees in firms nationwide can be approximately modeled by a lognormal
distribution. A fractional number of employees is allowed to account for part-time workers.
Suppose that
6.3
=
and
2
=
. For each of the following values of x, determine the number of
employees, such that p proportion of firms employ less than this number:
(a) 0.25
(b) 0.5
(c) 0.75
SOLUTION
Reserve Problems Chapter 4 Section 10 Problem 3
Derive the probability density function of a lognormal random variable from the derivative of the
cumulative distribution function.
SOLUTION
Let
( )
2
,XN
, then
X
Ye=
follows a lognormal distribution with mean
and variance
2
.
Reserve Problems Chapter 4 Section 10 Problem 4
Suppose that X has a lognormal distribution with parameters
10
=
and
216
=
. Determine the
following:
a)
( 2000)PX
b)
( 1500)PX
c) Value exceeded with probability 0.7
Reserve Problems Chapter 4 Section 10 Problem 5
The lifetime (in hours) of a semiconductor laser has a lognormal distribution with
8
=
and
1.5
=
. Determine the following in parts (a) and (b):
a) Probability the lifetime is less than 1000 hours
b) Probability the lifetime is less than 11000 hours given that it is more than 10000 hours
SOLUTION
a)
Reserve Problems Chapter 4 Section 11 Problem 1
News articles that link to related stories are widely used in Web marketing. With a large number
of daily visitors to a Web page, we model the proportion of daily visitors who click on a link to a
related story as approximately a continuous random variable with a beta distribution. The
parameters are
=
6 and
=
1.
(a) What is the mean and standard deviation of the proportion of visitors who click?
(b) What is the probability a proportion exceeds 0.58?
(c) What proportion is exceeded with probability 0.33?
(d) If 550 visitors view the page, what is the expected number of visitors who click on a link to a
related story?
SOLUTION
(a)
Let X denote the proportion of daily visitors who click on a link to a related story. Then, X is a
beta random variable with
=
3and
=
1, and its probability density function is
Reserve Problems Chapter 4 Section 11 Problem 2
Your exercise routine requires you to travel to the gym, complete your workout, and return
home. Suppose that the proportion of the total time for travel to the time at the gym follows a beta
distribution with
=
0.7 and
1
=
.
(a) What is the mean proportion of time required for travel to the gym?
(b) What is the probability that the proportion of time for travel to the gym exceeds 0.68?
SOLUTION
(a)
Let X denote the proportion of the total time of travel to the time in the gym. Then, X is a beta
Reserve Problems Chapter 4 Section 11 Problem 3
Suppose that X has a beta distribution with parameteres
1
=
and
4.2
=
. Determine the
following.
a)
( 0.25)PX=
b)
(0.50 )PX=
c)
( )
EX
==
d)
( )
2VX
==
SOLUTION
Reserve Supplemental Exercises Chapter 4 Problem 1
In applied genetics, the distribution of fitness effects (DFE) is used to determine the relative
abundance of mutations (harmful, neutral, or advantageous). According to a journal article [“The
distribution of fitness effects of new mutations” (2007, Nature Reviews Genetics, 8(8), pp. 610-
618)], the gamma distribution is most commonly used to model DFE. Suppose that a gamma
distribution with
1r=
,
2
=
is used as the DFE.
(a) What are the mean and the variance of the DFE?
(b) What value is exceeded with probability 0.50?
SOLUTION
(a) The probability density function for a gamma random variable X is
Reserve Supplemental Exercises Chapter 4 Problem 2
Many approaches in logistics involve a probability distribution of demand during the lead time to
set an order quantity and reorder point. A study [Computers & Operations Research “Inventory
management with log-normal demand per unit time” 2013, 40(7), pp. 1842-1851)] examined
optimal policies in a continuous review inventory management system. The customer demand
per unit time is assumed to follow an approximate lognormal distribution with
3
=
,
2
=
.
(a) Demand is measured in terms of the number of units. The demand cannot be exactly a
continuous distribution. Why?
(b) What is the probability that customer demand per unit time exceeds 2300?
(c) Determine the level of customer demand per unit time such that the probability of demand
less than this level is 0.04.
(d) Determine the mean of a customer demand per unit time. Determine the variance of a
customer demand per unit time.
SOLUTION
Reserve Supplemental Exercises Chapter 4 Problem 3
In a study of project management [“The use of buggers in project management: The trade-off
between stability and makespan” 2005, International Journal of Production Economics, 97(2),
pp. 227-240)], project executions were simulated using a beta distribution for activity durations.
Suppose that the simulated times to complete a task in this study followed a generalized beta
distribution from
10a=
to
20b=
days with
0.525
=
and
0.225
=
.
Determine the mean, variance and mode of the generalized beta distribution.
=
2
=
m
=
SOLUTION
Let a continuous random variable W denote the simulated time intervals needed to complete a
Reserve Supplemental Exercises Chapter 4 Problem 4
An article in American Heart Journal [“Long-term survival after successful in-hospital cardiac
arrest resuscitation” (2007, Vol 153(5), pp. 832-836)] examined long-term survival of patients
after in-hospital cardiac arrest. In this study, outcomes of 732 patients over 10 years were
analyzed. Suppose that the probability density function of patient mortality after in-hospital
cardiac arrest (in years) follows
( )
( )
0.5 3
0.5 x
f x e−−
=
for
3x
. Let a continuous random variable
X denote patient mortality after in-hospital cardiac arrest (in years).
(a) What are the mean and the variance of X?
(b) Determine the probability that X is more than nine years.
SOLUTION
(a) The mean and the variance are
Reserve Supplemental Exercises Chapter 4 Problem 5
Task requests to a data center are often modeled as a Poisson process. Assume that for a data
center, on average 100 tasks arrive per hour.
(a) Approximate the probability that more than 600 tasks arrive within 3 hours.
(b) Determine the mean of the time when the 200th task arrives.
Determine the standard deviation of the time when the 200th task arrives.
SOLUTION
(a) Let X denote the number of task requests to a data center that arrive within 3 hours. Then, X
Reserve Supplemental Exercises Chapter 4 Problem 6
In credit-card monitoring, 2% of the accounts with unusual activity in a month are actually due
to fraud. Suppose that 10,000 accounts with unusual activity are assumed independent.
Approximate the probability that more than 115 accounts are actually due to fraud.
SOLUTION
Reserve Supplemental Exercises Chapter 4 Problem 7
A city authority checks the hourly average weight of the traffic on a small bridge to determine if
more frequent maintenance should be conducted. Suppose X, the hourly average weight of the
traffic, is normally distributed with the mean of 30 tons and the standard deviation of 2.5 tons.
Determine the following:
(a)
( 37)PX
(b)
(28.5 32.5)PX
(c) Weight, which is exceeded with probability 0.99.
Reserve Supplemental Exercises Chapter 4 Problem 8
With an automated irrigation system, a plant grows to a height of 3.5 centimeters two weeks
after germination. Without an automated system, the height is normally distributed with mean
and standard deviation 2.5 and 0.5 centimeters, respectively.
a) With an automated irrigation system, a plant grows to a height of 3.5 centimeters two weeks
after germination.Without an automated system, the height is normally distributed with mean
and standard deviation 2.5 and 0.5 centimeters, respectively.
b) Do you think the automated irrigation system increases the plant height at two weeks after
germination?
SOLUTION
Reserve Supplemental Exercises Chapter 4 Problem 9
The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000
hours and a standard deviation of 600 hours.
a) What is the probability that a laser fails before 5700 hours?
b) What is the life in hours that 90% of the lasers exceed?
c) What should the mean life equal for 99% of the lasers to exceed 10000 hours before failure?
d) A product contains three lasers, and the product fails if any of the lasers fails. Assume the
lasers fail independently. What should the mean life equal for 99% of the products to exceed
10000 hours before failure?
SOLUTION
a)
Let X denote life:
600
