p151_exercises_4-11

# p151_exercises_4-11 - 37 The CPU of a personal computer has...

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Unformatted text preview: 37. The CPU of a personal computer has a lifetime that o nentially distributed with a mean lifetime of six years. - have owned this CPU for three years. *years? ‘3 Assume that your corporation has owned 10 CPUs for .5 three years, and assume that the CPUs fail independently. . What is the probability that at least one fails within the next three years? 38. Suppose that X has a lognormal distribution with ,I'neters 0 = 0 and (1)2 = 4. Determine the following: ‘5 P( 10 < X < 50) j,‘ The value for x such that P(X < x) = 0.05 7'. The mean and variance of X I 39. Suppose thatX has a lognormal distribution and that -" mean and variance of X are 50 and 4000, respectively. " 're the following: The parameters 0 and (92 of the lognormal distribution , The probability thatX is less than 150 40. Asbestos ﬁbers in a dust sample are identiﬁed by an '«tron microscope after sample preparation. Suppose that 1" number of ﬁbers is a Poisson random variable and the . number of ﬁbers per squared centimeter of surface dust 00. A sample of 800 square centimeters of dust is analyzed. e a particular grid cell under the microscope represents .‘60,000 of the sample. 5 What is the probability that at least one ﬁber is visible in -‘ the grid cell? ‘ What is the mean of the number of grid cells that need to .-' be viewed to observe 10 that contain ﬁbers? is What is the standard deviation of the number of grid cells . that need to be viewed to observe 10 that contain ﬁbers? .- 41. Without an automated irrigation system, the height (plants two weeks after germination is normally distributed in a mean of 2.5 centimeters and a standard deviation of 0.5 ‘ timeters. What is the probability that a plant’s height is greater than f 2.25 centimeters? ) What is the probability that a plant’s height is between 2.0 ‘ and 3.0 centimeters? , What height is exceeded by 90% of the plants? .3 142. Continuation of Exercise 4-141. With an automated ; ' ation system, a plant grows to a height of 3.5 centimeters rs weeks after germination. ) What is the probability of obtaining a plant of this height or greater ﬁ'om the distribution of heights in Exercise 4-135. ) Do you think the automated irrigation system increases the plant height at two weeks after germination? '143. The thickness of a laminated covering for a wood ,n ace is normally distributed with a mean of 5 millimeters ‘rd a standard deviation of 0.2 millimeter. ) What is the probability that a covering thickness is greater than 5.5 millimeters? What is the probability that the CPU fails in the next three. 441 LOGNORMAL DISTRIBUTION 1 5 1 (b) If the speciﬁcations require the thickness to be between 4.5 and 5.5 millimeters, what proportion of coverings do not meet speciﬁcations? (c) The covering thickness of 95% of samples is below what value? 4—144. The diameter of the dot produced by a printer is nor- mally distributed with a mean diameter of 0.002 inch. Suppose that the speciﬁcations require the dot diameter to be between 0.0014 and 0.0026 inch. If the probability that a dot meets speciﬁcations is to be 0.9973, what standard deviation is needed? 4—145. Continuation of Exercise 4-144. Assume that the stande deviation of the size of a dot is 0.0004 inch. If the probability that a dot meets speciﬁcations is to be 0.9973, what speciﬁcations are needed? Assume that the speciﬁca- tions are to be chosen symmetrically around the mean of 0.002. 4— 146. The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a stan- dard deviation of 600 hours. (a) What is the probability that a laser fails before 5,800 hours? (b) What is the life in hours that 90% of the lasers exceed? (c) What should the mean life equal in order for 99% of the lasers to exceed 10,000 hours before failure? (d) A product contains three lasers, and the product fails if any of the lasers fails. Assume the lasers fail independ- ently. What should the mean life equal in order for 99% of the products to exceed 10,000 hours before failure? 4—147. Continuation of Exercise 4-146. Rework parts (a) and (b). Assume that the lifetime is an exponential random variable with the same mean. 4-148. Continuation of Exercise 4-146. Rework parts (a) and (b). Assume that the lifetime is a lognormal random vari— able with the same mean and standard deviation. 4’149. A square inch of carpeting contains 50 carpet ﬁbers. The probability of a damaged ﬁber is 0.0001. Assume the damaged ﬁbers occur independently. (a) Approximate the probability of one or more damaged ﬁbers in 1 square yard of carpeting. (b) Approximate the probability of four or more damaged ﬁbers in 1 square yard of carpeting. 4—150. An airline makes 200 reservations for a ﬂight that holds 185 passengers. The probability that a passenger arrives for the ﬂight is 0.9 and the passengers are assumed to be inde- pendent. (3) Approximate the probability that all the passengers that arrive can be seated. (b) Approximate the probability that there are empty seats. (c) Approximate the number of reservations that the airline should make so that the probability that everyone who arrives can be seated is 0.95. [Hintz Successively try values for the number of reservations] ...
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