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Problem Set No. 3 (ENGSTAT) 1. A study of five-year trends in the logistics information systems of industries found that the greatest computerization advances were in transportation (Industrial Engineering, July 1990). Currently, 90% of all industries contain shipping open order files in their computerized data base. In a random sample of 10 industries, let y equal the number that include the shipping open order files in their computerized data base. a. Verify that the probability distribution of y can be modeled using the binomial distribution b. Find P(y=7) c. Find P(y>5) d. Find the mean of y. Interpret the results. 2. The negative binomial distribution was used to model the distribution of parasites (tapeworms) found in several species of Mediterranean fish (Journal of Fish Biology, Aug. 1990). Assume the event of interest is whether or not a parasite is found in the digestive tract of brill fish, and let y be the number of brill that must be sampled until a parasite infection is found. The researchers estimate the probability of an infected fish at .544. Use this information to estimate the following probabilities: a. P (y=3); b. P (y ≤ 2); c. P (y > 2) 3. Two of the five mechanical engineers employed by the country sanitation department have experience in the design of steam turbine power plants. You have been instructed to choose randomly two of the five engineers to work on a project for a new power plant. a. What is the probability that you will choose the engineers with the experience in the design of steam turbine power plants? b. What is the probability that you will choose at least one of the engineers with such experience? 4. A study of vehicle flow characteristics on acceleration lanes (i.e. merging ramps) at a major freeway in Israel found that one out of every six vehicles use less than one-third of the acceleration lane before merging into traffic (Journal of Transportation Engineering, Nov. 1985). Suppose we monitor the location of the merge for the next five vehicles that enter the acceleration lane. a. What is the probability that none of the vehicles will use less than one-third of the acceleration lane? b. What is the probability that exactly two of the vehicles will use less than one-third of the acceleration lane? 5. When radar was first introduced during World War II, it was very difficult for an operator manning the screen to distinguish a static interference blip from an actual enemy aircraft blip. Although the operator did not want to sound an alarm needlessly, failure to alert the defenses could have serious consequences. Records indicate that 60% of all observed blips represented enemy aircraft. Suppose that during a particular siege there were five blips spotted on the screen at different points in time and the radio operator alerted the defenses on each occasion. Assume that the events are independent and compute the probability of each of the following events. a. Radar operator made the correct decision on all five occasions b. Radar operator made the correct decision on at least five occasions c. Radar operator was incorrect all five times (and therefore sounded five false alarms) 6. Researchers at the University of California-Berkeley have designed, built, and tested a switchedcapacitor circuit for generating random signals (International Journal of Circuit Theory and Applications, May-June 1990). The circuit’s trajectory was shown to be uniformly distributed on the interval (0, 1). a. Give the mean and variance of the circuit’s trajectory b. Compute the probability that the trajectory falls between .2 and .4 7. Use Table 4 of Appendix II to find the following probabilities for a standard normal random variable: a. P(.5 < z < 1.5) b. P(-1.75 < z < -2.8) c. P(-2.32 < z < .11) d. P(z > .27) e. P(z < -1.33) f. P(z < 1.71) 8. Find the value of the standard normal random variable z, call it z0, such that: a. P(z > z0) = .05 b. P(z > z0) = .025 c. P(z > z0) = 80 d. P(z > z0) = .0013 e. P(z > z0) = .97 f. P(z > z0) = .5596 9. Steel used for water pipelines is often coated on the inside with cement mortar to prevent corrosion. In a study of the mortar coatings of a pipeline used in a water transmission project in California (Transportation Engineering Journal, Nov. 1979), the mortar thickness was specified to be 7/16 inch. A very large number of thickness measurements produced a mean equal to .635 inch and a standard deviation equal to .082 inch. If the thickness measurements were normally distributed, approximately what percentage was less than 7/16 inch? 10. The distribution of the demand (in number of units per unit time) for a product can often be approximated by a normal probability distribution. For example, a communication cable company has determined that the number of push-button terminal switches demanded daily has a normal distribution with mean 200 and standard deviation 50. a. On what percentage of days will the demand be less than 90 switches? b. On what percentage of days will the demand fall between 225 and 275 switches? c. Based on cost considerations, the company has determined that its best strategy is to produce a sufficient number of switches so that it will fully supply demand on 94% of all days. How many terminal switches should the company produce per day? 11. Researchers have discovered that the maximum flood level (in millions of cubic feet per second over a 4-year period for the Susquehanna River in Harrisburg, Pennsylvania, follows approximately a gamma distribution with = 3 and = .07 (Journal of Quality Technology, Jan. 1986). a. Find the mean and variance of the maximum flood level over a 4-year period for the Susquehanna River 12. 13. 14. 15. 16. b. The researchers arrived at their conclusion about the maximum flood level distribution by observing maximum flood levels over 20 4-year periods from 1890-1969. Suppose that over the 4-year period 1982-1985 the maximum flood level was observed to be .60 million cubic feet per second. Would you expect to observe a value this high from a gamma distribution with = 3 and = .07? What can you infer about the maximum flood level distribution for the 4-year period 1982-1985? In finding and correcting errors in a computer program (debugging) and determining the program’s reliability, computer software experts have noted the importance of the distribution of the time until the next program error is found. Suppose that this random variable has a gamma distribution with parameter = 1. One computer programmer believes that the mean time between finding program errors is = 24 days. Suppose that a programming error is found today. Assuming that = 24, find the probability that it will take at least 60 days to discover the next programming error Wind models are used in engineering design for wind energy analysis and design limit wind speeds. A widely used model of wind speed y (in miles per second) is the Weibull density function with parameters = 1 and = v/2, where v (the characteristic speed) is the 63.21 percentile of the wind speed distribution (Atmospheric Environment, Vol. 18, No. 10, 1984). At a particular site in Great Britain, the characteristic wind speed is known to be v = 11.3 miles per second. Use the Weibull wind model to find: a. E(y) and a2 b. The probability that wind speed y is less than 6 miles per second c. The probability that wind speed y is greater than 10 miles per second Japanese electrical engineers have developed a sophisticated radar system called the moving target detector (MTD), designed to reject ground clutter, rain clutter, birds, and other interference (IEE Proceedings, Aug. 1984). The researchers show that the magnitude y of the Doppler frequency of a radar-received signal obeys a Weibull distribution with parameters = 2 and . a. Find E(y) b. Find 2 The metropolitan airport commission is considering the establishment of limitations on the extent of noise pollution around a local airport. At the present time the noise level per jet takeoff in one neighborhood near the airport is approximately normally distributed with a mean of 100 decibels and a standard deviation of 6 decibels. a. What is the probability that a randomly selected jet will generate a noise level greater than 108 decibels? b. What is the probability that a randomly selected jet will generate a noise level of exactly 100 decibels? c. Suppose that a regulation is passed that requires jet noise in the neighborhood to be lower than 105 decibels 95% of the time. Assuming the standard deviation of the noise distribution remains the same, how much will the mean noise level have to be lowered to comply with the regulation? The length of time between breakdowns of an essential piece of equipment is an important factor in deciding on the amount of auxiliary equipment needed to assure continuous service. A machine room foreman believes the time between breakdowns of a particular electrical generator is best approximated by an exponential distribution with mean equal to 10 days. a. What is the standard deviation of this exponential distribution? b. Assuming that the foreman has correctly characterized the distribution for the time between breakdowns and that the generator broke down today, what is the probability that the generator will break down again within the next 14 days? c. What is the probability that the generator will operate for more than 20 days without a breakdown?