ISE 261 HOMEWORK FOUR Due Date: Tuesday 3/13
1.The engineering department of a steel manufacturer is analyzingthe 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 RSVP-linked 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 anuniform 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 diameterμ = 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 thatL 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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