5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES

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B D B

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Figure 5-20 Figure for the U-shaped component.

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(b) What is the probability that the width of the casing minus the width of the door exceeds 1 4 inch (c) What is the probability that the door does not t in the casing 5-92. A U-shaped component is to be formed from the three parts A, B, and C. The picture is shown in Fig. 5-20. The length of A is normally distributed with a mean of 10 millimeters and a standard deviation of 0.1 millimeter. The thickness of parts B and C is normally distributed with a mean of 2 millimeters and a standard deviation of 0.05 millimeter. Assume all dimensions are independent. (a) Determine the mean and standard deviation of the length of the gap D. (b) What is the probability that the gap D is less than 5.9 millimeters 5-93. Soft-drink cans are lled by an automated lling machine and the standard deviation is 0.5 uid ounce. Assume that the ll volumes of the cans are independent, normal random variables. (a) What is the standard deviation of the average ll volume of 100 cans (b) If the mean ll volume is 12.1 ounces, what is the probability that the average ll volume of the 100 cans is below 12 uid ounces (c) What should the mean ll volume equal so that the probability that the average of 100 cans is below 12 uid ounces is 0.005 (d) If the mean ll volume is 12.1 uid ounces, what should the standard deviation of ll volume equal so that the probability that the average of 100 cans is below 12 uid ounces is 0.005 (e) Determine the number of cans that need to be measured such that the probability that the average ll volume is less than 12 uid ounces is 0.01. 5-94. The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Assume that the thickness is normally distributed and that the thicknesses of different wafers are independent. (a) Determine the probability that the average thickness of 10 wafers is either greater than 11 or less than 9 micrometers.

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(b) Determine the number of wafers that needs to be measured such that the probability that the average thickness exceeds 11 micrometers is 0.01. (c) If the mean thickness is 10 micrometers, what should the standard deviation of thickness equal so that the probability that the average of 10 wafers is either greater than 11 or less than 9 micrometers is 0.001 5-95. Assume that the weights of individuals are independent and normally distributed with a mean of 160 pounds and a standard deviation of 30 pounds. Suppose that 25 people squeeze into an elevator that is designed to hold 4300 pounds. (a) What is the probability that the load (total weight) exceeds the design limit (b) What design limit is exceeded by 25 occupants with probability 0.0001

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5-8 FUNCTIONS OF RANDOM VARIABLES (CD ONLY) 5-9 MOMENT GENERATING FUNCTION (CD ONLY) 5-10 CHEBYSHEV S INEQUALITY (CD ONLY) Supplemental Exercises

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5-96. Show that the following function satis es the properties of a joint probability mass function: x 0 0 1 1 2 y 0 1 0 1 2 f(x, y) 1 1 1 1 1 4 8 8 4 4

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5-97. Continuation of Exercise 5-96. Determine the following probabilities: (a) P1X 0.5, Y 1.52 (b) P1X 12 (c) P1X 1.52 (d) P1X 0.5, Y 1.52 (e) Determine E(X ), E(Y ), V(X ), and V(Y). 5-98. Continuation of Exercise 5-96. Determine the following: (a) Marginal probability distribution of the random variable X (b) Conditional probability distribution of Y given that X 1 (c) E1Y 0 X 12 (d) Are X and Y independent Why or why not (e) Calculate the correlation between X and Y. 5-99. The percentage of people given an antirheumatoid medication who suffer severe, moderate, or minor side effects

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