Download ISE 261 HOMEWORK FOUR Due Date: Thursday 3/22/2012 1. The

ISE 261
HOMEWORK FOUR
Due Date: Thursday 3/22/2012
1. The engineering department of a steel manufacturer is analyzing the company’s rolling
machine that’s produces sheets of steel with varying thickness. The thickness is found to be a
uniform random variable with values between 150 and 200 millimeters. Any sheets less than
160 millimeters thick must be scrapped, since they are unacceptable to buyers. Find the
probability distribution for thickness f(x), and then calculate the mean and standard deviation of
the thickness of the sheets produced by this machine.
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2. The amount of time (in minutes) that a commuter train is late is a continuous random variable
with the probability density function listed below. Find the mean and variance of the amount of
time in minutes the train is late. (Note: A negative time value means that the train is early).
f(x) = 3(25 – x2) / 500 for –5 < x < +5
0 elsewhere
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3. Resource reservation protocol (RSVP) was originally designed to establish signaling links for
stationary networks. In Mobile Networks and Applications (Dec. 2003), RSVP was applied to
mobile wireless technology (e.g., a PC notebook with wireless LAN card for Internet access). A
simulation study revealed that the transmission delay (measured in milliseconds) of an RSVPlinked wireless device has an approximate normal distribution with mean μ = 48.5 milliseconds
and σ = 8.5 milliseconds. What is the probability that the transmission delay is between 40 and
60 milliseconds?
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4. The reliability of a piece of equipment is frequently defined to be the probability, P, that the
equipment performs its intended function successfully for a given period of time under specific
conditions. Because P varies from one point in time to another, some reliability analysts treat P
as if it were a random variable. Suppose an analyst characterizes the uncertainty about the
reliability of an articulating robot used in an automobile assembly line using the pdf. listed
below. Then, after careful investigation, the analyst discovers that P is definitely between 0.90
and 0.95, but that there is complete uncertainty about where it lies between these values.
Describe the pdf. the analyst should now use to describe P.
f(p) = 1 for 0 < p < 1
0 otherwise
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5. A manufacturing company has developed a fuel-efficient machine that combines pressure
washing with steam cleaning. It is designed to deliver 7 gallons of cleaner per minute at 1,000
pounds per square inch of pressure washing. In fact, it delivers an amount at random anywhere
between 6.5 and 7.5 gallons per minute. Assume that the RV X, the amount of cleaner
delivered, is an uniform RV with probability density f(x) = 1 for 6.5 < x < 7.5. What is the
probability that more than 7.2 gallons of cleaner are dispensed per minute?
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6. The Transactions of the ASME recently presented a model for predicting daily natural gas
consumption in urban areas. A key component of the model is the distribution of daily
temperatures in the area. Based on daily July temperatures collected in Buenos Aires,
Argentina, from 1944 to 2000, researchers demonstrated that the daily July temperature is
Normally distributed with mean μ = 11o C and σ = 3.1oC. Suppose you want to use temperature
to predict natural gas consumption on future July day in Buenos Aires. Give a temperature
value that is exceeded on only 5% of the July days in Buenos Aires.
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7. Ecological Applications published a study on the development of forests following wildfires
in the Pacific Northwest. One variable of interest to the researcher was tree diameter at breast
height 110 years after the fire. The population of Douglas fir trees was shown to have an
approximately normal distribution with mean tree diameter μ = 50 centimeters (cm) and σ = 12
cm. Find the diameter, d, such that 30% of the Douglas fir trees in the population have
diameters that exceed d.
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8. The length of time T (in minutes) required to generate a human reaction to tear gas formula A
has a gamma distribution with α = 2 and β = 2 minutes. The distribution of formula B is also
gamma, but with α = 1 and β = 4 minutes. Which tear gas has a higher probability of generating
a human reaction in less than 4 minutes?
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9. Based on data collected from metal shredders across the nation, the amount L of extractable
lead in metal shredder residue has an approximate exponential distribution with mean μ = 2.5
milligrams per liter. What is the probability that L is greater than 2 milligrams per liter?
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10. The temperature reading from a thermocouple placed in a constant-temperature medium is
normally distributed with mean μ, the actual temperature of the medium, and standard deviation
σ. What would the value of σ have to be to ensure that 80% of all readings are within 0.1O of μ?
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11. The length of time (in months after maintenance) until failure of a bank’s surveillance
television equipment has a Weibull distribution with α = 2 and β =60 months. If the bank wants
the probability of a breakdown before the next scheduled maintenance to be 0.05, how
frequently should the equipment receive periodic maintenance?
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12. Data collected over time on the utilization of a computer core (as a proportion of the total
capacity) were found to possess a relative frequency distribution that could be approximated by
a Beta density function with α = 2 and β = 4. Find the probability that the proportion of the core
being used at any particular time will be less than 0.20.
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13. Based on data from a dart-throwing experiment, the article “Shooting Stars” in Chance
proposed that the horizontal and vertical errors from aiming at a point target should be
independent of one another, each with a normal distribution having mean μ = 0 and σ = 20mm.
(It can then be shown that the pdf of the squared distance between the point target and the
landing point T, divided by the variance, follows a Chi-Squared probability density function
with υ = 2 degrees of freedom). What is the probability that a dart will land within 54.325 mm
of the target?
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14. The intensity of one sound can be compared to that of another of the same frequency by
taking the ratio of their powers. When the ratio is 10, the difference in intensity of the sounds is
said to be one bel, a unit named in honor of Alexander Graham Bell. The unit in general use is
the decibel (dB), equal to 0.1 bel. Decibels are also used to express the ratio of the magnitudes
of two electric voltages or currents. In this usage one dB equals 20 times the common logarithm
of the ratio. It can be shown that rate-data often follow a lognormal distribution. Average power
usage (dB per hour) for a particular company is studied and is known to have a lognormal
distribution with parameters μln=4 and σln = 2 (dB per hour). What is the probability that the
company uses more than 270 dB during any particular hour?
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