ISE 261 HOMEWORK FIVE Due Date:Tuesday 4/10/2018
1.An article in the Journal of Engineering for Industry reports on the researchbeing prepared on the friction that occurs in the paper-feeding of a photocopier. The coefficient of friction is a proportion that measures the degree of friction between two adjacent sheets of paper in the feeder stack. The article described a system that utilized two interrelated feed paper separators. The joint density of X and Y, the friction coefficients of the two machines is shown below. Find the probability that both friction coefficients exceed 0.80.
f(x,y) = xyif 0 ≤ x ≤ 1; 0 ≤ y ≤ 1
(2 – x)yif 1 ≤ x ≤ 2; 0 ≤ y ≤ 1
(2 – y)xif 0 ≤ x ≤ 1; 1 ≤ y ≤ 2
(2 – x)(2 – y) if 1 ≤ x ≤ 2; 1 ≤ y ≤ 2
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2.Fruits are important sources of dietary fibre and vitamins (especially vitamin C). From a sack of fruit containing three oranges, two apples, and three bananas, a random sample of four pieces of fruit is selected. If X is the number of whole oranges and Y is the number of whole apples in the sample, find the joint probability distribution and marginal probability distributions of X and Y.
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3.The amount of kerosene, in thousands of liters, in a tank at the beginning of any day is a random amount Y from which a random amount X is sold during that day. Suppose that the tank is not resupplied during the day so that x ≤ y, and assume that the joint density function of these variables is f(x,y) = 2 for 0 < x < y < 1. Determine if X and Y are independent.
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4.Referring to problem #3, find the probability that the amount of kerosene sold is between 0.25and 0.50 thousands liters given that the amount in the tank at the start of the day is 0.75 thousands liters. (Hint: f(x|y) = f(x,y) / fy(y) ).
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5.Eight individuals, including A and B, take seats around a circular table in a completely random fashion. Suppose the seats are numbered 1, …, 8. Let X = A’s seat number and Y = B’s seat number. If A sends a written message around the table to B in the direction in which they are closest, how many individuals (including A and B) would you expect to handle the message?
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6.A surveyor wishes to lay out a square region with each side having length L. However, because of measurement error, he instead lays out a rectangle in which the north-south sides both have length X and the east-west sides both have length Y. Suppose that the joint pmf of X and Y is f(x,y) = 1 / (4L2) for 0 ≤ x ≤ 2L and 0 ≤ y ≤ 2L. What is the expected area of the resulting rectangle?
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7.Find the covariance of X and Y in problem #2.
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8.The fraction X of male runners and the fraction Y of female runners who compete in marathon races are described by the joint density function f(x,y) = 8xy for 0 ≤ y ≤ x ≤ 1. Find the covariance of X and Y.
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9.Each front tire on a hybrid is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable: X for the right tire and Y for the left tire with joint pdf. f(x,y) = k(x2 + y2) for 20 ≤ x ≤ 30 and 20 ≤ y ≤ 30where k = 3/380,000. Find the correlation coefficient ρ for X and Y.
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10.Civil engineers responsible for the design and maintenance of aircraft pavements traditionally use pavement-quality concrete. A study was conducted to access the suitability of concrete blocks as a surface for aircraft pavements. The original pavement-quality concrete of the western end of a runway was overlaid with 80-mm-thick concrete blocks. A series of plate-bearing tests was carried out to determine the load classification number (LCN), a measure of breaking strength, of the surface. Let RV X represent the mean LCN of a sample of 25 concrete block sections on the western end of the runway. Prior to resurfacing, the mean LCN of the original payment-quality concrete of the western end of the runway was known to be 60 with standard deviation σ = 10. If the mean strength of the new concrete block surface is no different from that of the original surface, find the probability that the sample mean LCN of the 25 concrete block sections exceeds 65.
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11.A rivet is a permanent mechanical fastener. Before being installed a rivet consists of a smooth cylindrical shaft with a head on one end. The end opposite the head is called the buck-tail. On installation the rivet is placed in a punched or pre-drilled hole, and the tail is upset, or bucked (i.e. deformed), so that it expands to about 1.5 times the original shaft diameter, holding the rivet in place.The breaking strength of a special rivet has a mean value of 10,000 psi and a standard deviation of 500 psi. What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 9,900 and 10,200?
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12. An important manufacturing process produces cylindrical components for the automotive industry. It is important that the process produces parts having a mean of 5.0 millimeters. The engineer involved conjectures that the population mean is 5.0 millimeters. An experiment is conducted in which 100 parts produced by the process are selected randomly and the diameter measured on each. It is known that the population standard deviation σ = 0.10. The experiment indicates a sample average diameter x-bar = 5.027 millimeters. Does this sample information appear to support the engineer’s conjecture? (Prove your answer).
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THE END