In an accelerator center, an experiment needs a 1.41-cmthick aluminum cylinder

Suppose that for Determine the following: (a) (b) (c) (d) (e) (f) Determine x such that (g) Determine x such that

Suppose that Determine the following: (a) (b) (c) (d)

Suppose that Determine the following: (a) (b) (c) (d) (e) Determine x such that

The diameter of a particle of contamination (in micrometers) is modeled with the probability density function for Determine the following: (a) (b) (c) (d) (e) Determine x such that

Suppose that for Determine the following probabilities: (a) (b) (c) (d) (e)

Suppose that Determine the following: (a) (b) (c) (d) (e) Determine x such that

Suppose that for Determine the following: (a) (b) (c) P10.5 X 0.52 (d) P1 (f) Determine x such that

The probability density function of the time to failure of an electronic component in a copier (in hours) is f(x) for Determine the probability that (a) A component lasts more than 3000 hours before failure. (b) A component fails in the interval from 1000 to 2000 hours. (c) A component fails before 1000 hours. (d) Determine the number of hours at which 10% of all components have failed.

The probability density function of the net weight in pounds of a packaged chemical herbicide is for pounds. (a) Determine the probability that a package weighs more than 50 pounds. (b) How much chemical is contained in 90% of all packages?

The probability density function of the length of a cutting blade is for millimeters. Determine the following: (a) (b) (c) If the specifications for this process are from 74.7 to 75.3 millimeters, what proportion of blades meets specifications?

The probability density function of the length of a metal rod is for 2.3  x  2.8 meters. (a) If the specifications for this process are from 2.25 to 2.75 meters, what proportion of rods fail to meet the specifications? (b) Assume that the probability density function is for an interval of length 0.5 meters. Over what value should the density be centered to achieve the greatest proportion of bars within specifications?

If X is a continuous random variable, argue that P(x1 X x2) P(x1 X x2) P(x1 X  x2) P(x1  X  x2). f 1x2

Suppose the cumulative distribution function of the random variable X is Determine the following: (a) (b) (c) (d)

Suppose the cumulative distribution function of the random variable X is Determine the following: (a) (b) (c) (d)

Determine the cumulative distribution function for the distribution in Exercise 4-1.

Determine the cumulative distribution function for the distribution in Exercise 4-2.

Determine the cumulative distribution function for the distribution in Exercise 4-3.

Determine the cumulative distribution function for the distribution in Exercise 4-4.

Determine the cumulative distribution function for the distribution in Exercise 4-5.

Determine the cumulative distribution function for the distribution in Exercise 4-8. Use the cumulative distribution function to determine the probability that a component lasts more than 3000 hours before failure.

Determine the cumulative distribution function for the distribution in Exercise 4-11. Use the cumulative distribution function to determine the probability that a length exceeds 2.7 meters.

The probability density function of the time customers arrive at a terminal (in minutes after 8:00 A.M.) is for 0  x. Determine the probability that (a) The first customer arrives by 9:00 A.M. (b) The first customer arrives between 8:15 A.M. and 8:30 A.M. (c) Two or more customers arrive before 8:40 A.M. among five who arrive at the terminal. Assume cus tomer arrivals are independent. (d) Determine the cumulative distribution function and use the cumulative distribution function to determine the probability that the first customer arrives between 8:15 A.M. and 8:30 A.M.

The gap width is an important property of a magnetic recording head. In coded units, if the width is a continuous random variable over the range from 0  x  2 with f(x) 0.5x, determine the cumulative distribution function of the gap width.

Determine the probability density function for each of the following cumulative distribution functions.F1x2 1  e2x x  0 4

Determine the probability density function for each of the following cumulative distribution functions.F1x2 0 x  0 0.2x 0 x  4 0.04x 0.64 4 x  9 1 9 x JWC

Determine the probability density function for each of the following cumulative distribution functions.F1x2 0 x  2 0.25x 0.5 2 x  1 0.5x 0.25 1 x  1.5 1 1.5 x F1x

Suppose for Determine the mean and variance of X.

Suppose for Determine the mean and variance of X.

Suppose for Determine the mean and variance of X.

Suppose that for Determine the mean and variance of x.

Determine the mean and variance of the random variable in Exercise 4-2.

Determine the mean and variance of the random variable in Exercise 4-1.

Suppose that contamination particle size (in micrometers) can be modeled as for Determine the mean of X.

Suppose the probability density function of the length of computer cables is f(x) 0.1 from 1200 to 1210 millimeters. (a) Determine the mean and standard deviation of the cable length. (b) If the length specifications are 1195  x  1205 millimeters, what proportion of cables are within specifications?

The thickness of a conductive coating in micrometers has a density function of 600x2 for 100 m  x  120 m. (a) Determine the mean and variance of the coating thickness. (b) If the coating costs $0.50 per micrometer of thickness on each part, what is the average cost of the coating per part?

The probability density function of the weight of packages delivered by a post office is for 1  x  70 pounds. (a) Determine the mean and variance of weight. (b) If the shipping cost is $2.50 per pound, what is the average shipping cost of a package? (c) Determine the probability that the weight of a package exceeds 50 pounds.

Integration by parts is required. The probability density function for the diameter of a drilled hole in millimeters is for mm. Although the target diameter is 5 millimeters, vibrations, tool wear, and other nuisances produce diameters larger than 5 millimeters. (a) Determine the mean and variance of the diameter of the holes. (b) Determine the probability that a diameter exceeds 5.1 millimeters.

Suppose X has a continuous uniform distribution over the interval [1.5, 5.5]. (a) Determine the mean, variance, and standard deviation of X. (b) What is ? (c) Determine the cumulative distribution function.

Suppose X has a continuous uniform distribution over the interval (a) Determine the mean, variance, and standard deviation of X. (b) Determine the value for x such that P(x  X  x) 0.90. (c) Determine the cumulative distribution function.

The net weight in pounds of a packaged chemical herbicide is uniform for pounds. (a) Determine the mean and variance of the weight of packages. (b) Determine the cumulative distribution function of the weight of packages. (c) Determine

The thickness of a flange on an aircraft component is uniformly distributed between 0.95 and 1.05 millimeters. (a) Determine the cumulative distribution function of flange thickness. (b) Determine the proportion of flanges that exceeds 1.02 millimeters. (c) What thickness is exceeded by 90% of the flanges? (d) Determine the mean and variance of flange thickness.

Suppose the time it takes a data collection operator to fill out an electronic form for a database is uniformly between 1.5 and 2.2 minutes. (a) What is the mean and variance of the time it takes an operator to fill out the form? (b) What is the probability that it will take less than two minutes to fill out the form? (c) Determine the cumulative distribution function of the time it takes to fill out the form.

The thickness of photoresist applied to wafers in semiconductor manufacturing at a particular location on the wafer is uniformly distributed between 0.2050 and 0.2150 micrometers. (a) Determine the cumulative distribution function of photoresist thickness. (b) Determine the proportion of wafers that exceeds 0.2125 micrometers in photoresist thickness. (c) What thickness is exceeded by 10% of the wafers? (d) Determine the mean and variance of photoresist thickness.

An adult can lose or gain two pounds of water in the course of a day. Assume that the changes in water weight are uniformly distributed between minus two and plus two pounds in a day. What is the standard deviation of your weight over a day?

A show is scheduled to start at 9:00 A.M., 9:30 A.M., and 10:00 A.M. Once the show starts, the gate will be closed. A visitor will arrive at the gate at a time uniformly distributed between 8:30 A.M. and 10:00 A.M. Determine (a) The cumulative distribution function of the time (in minutes) between arrival and 8:30 A.M. (b) The mean and variance of the distribution in the previous part. (c) The probability that a visitor waits less than 10 minutes for a show. (d) The probability that a visitor waits more than 20 minutes for a show.

The volume of a shampoo filled into a container is uniformly distributed between 374 and 380 milliliters. (a) What are the mean and standard deviation of the volume of shampoo? (b) What is the probability that the container is filled with less than the advertised target of 375 milliliters? (c) What is the volume of shampoo that is exceeded by 95% of the containers? (d) Every milliliter of shampoo costs the producer $0.002. Any more shampoo in the container than 375 milliliters is an extra cost to the producer. What is the mean extra cost?

An e-mail message will arrive at a time uniformly distributed between 9:00 A.M. and 11:00 A.M. You check e-mail at 9:15 A.M. and every 30 minutes afterward. (a) What is the standard deviation of arrival time (in minutes)? (b) What is the probability that the message arrives less than 10 minutes before you view it? (c) What is the probability that the message arrives more than 15 minutes before you view it?

Measurement error that is continuous and uniformly distributed from 3 to 3 millivolts is added to the true voltage of a circuit. Then the measurement is rounded to the nearest millivolt so that it becomes discrete. Suppose that the true voltage is 250 millivolts. (a) What is the probability mass function of the measured voltage? (b) What is the mean and variance of the measured voltage?

Use Appendix Table III to determine the following probabilities for the standard normal random variable Z: (a) P(Z  1.32) (b) P(Z  3.0) (c) P(Z  1.45) (d) P(Z  2.15) (e)P( 2.34 Z 1.76)

Use Appendix Table III to determine the following probabilities for the standard normal random variable Z: (a) P(1  Z  1) (b) P(2  Z  2) (c) P(3  Z  3) (d) P(Z  3) (e)P(0 Z 1)

Assume Z has a standard normal distribution. Use Appendix Table III to determine the value for z that solves each of the following: (a) P( Z  z) 0.9 (b) P(Z  z) 0.5 (c) P( Z  z) 0.1 (d) P(Z  z) 0.9 (e) P(1.24  Z  z) 0.8 4-5

Assume Z has a standard normal distribution. Use Appendix Table III to determine the value for z that solves each of the following: (a) P(z  Z  z) 0.95 (b) P(z  Z  z) 0.99 (c) P(z  Z  z) 0.68 (d) P(z  Z  z) 0.9973 4-5

Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the following: (a) P(X  13) (b) P(X  9) (c) P(6  X  14) (d) P(2  X  4) (e) P(2  X  8)

Assume X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the value for x that solves each of the following: (a) P(X  x) 0.5 (b) P(X  x) 0.95 (c) P(x  X  10) 0.2 (d) P(x  X  10  x) 0.95 (e) P(x  X  10  x) 0.99 4-55

Assume X is normally distributed with a mean of 5 and a standard deviation of 4. Determine the following: (a) P(X  11) (b) P(X  0) (c) P(3  X  7) (d) P(2  X  9) (e) P(2  X  8)

Assume X is normally distributed with a mean of 5 and a standard deviation of 4. Determine the value for x that solves each of the following:(a) P(X x) 0.5 (b) P(X x) 0.95 (c) P(x X 9) 0.2 (d) P(3 X x) 0.95 (e) P( x X 5 x) 0.99 4-57

The compressive strength of samples of cement can be modeled by a normal distribution with a mean of 6000 kilograms per square centimeter and a standard deviation of 100 kilograms per square centimeter. (a) What is the probability that a samples strength is less than 6250 Kg/cm2 ? (b) What is the probability that a samples strength is between 5800 and 5900 Kg/cm2 ? (c) What strength is exceeded by 95% of the samples?

The time until recharge for a battery in a laptop computer under common conditions is normally distributed with a mean of 260 minutes and a standard deviation of 50 minutes. (a) What is the probability that a battery lasts more than four hours? (b) What are the quartiles (the 25% and 75% values) of battery life? (c) What value of life in minutes is exceeded with 95% probability?

An article in Knee Surgery Sports Traumatol Arthrosc [Effect of Provider Volume on Resource Utilization for Surgical Procedures (2005, Vol. 13, pp. 273279)] showed a mean time of 129 minutes and a standard deviation of 14 minutes for ACL reconstruction surgery at high-volume hospitals (with more than 300 such surgeries per year). (a) What is the probability that your ACL surgery at a highvolume hospital requires a time more than two standard deviations above the mean? (b) What is the probability that your ACL surgery at a highvolume hospital is completed in less than 100 minutes? (c) The probability of a completed ACL surgery at a highvolume hospital is equal to 95% at what time? (d) If your surgery requires 199 minutes, what do you conclude about the volume of such surgeries at your hospital? Explain.

Cholesterol is a fatty substance that is an important part of the outer lining (membrane) of cells in the body of animals. Its normal range for an adult is 120240 mg/dl. The Food and Nutrition Institute of the Philippines found that the total cholesterol level for Filipino adults has a mean of 159.2 mg/dl and 84.1% of adults have a cholesterol level below 200 mg/dl (http://www.fnri.dost.gov.ph/). Suppose that the total cholesterol level is normally distributed. (a) Determine the standard deviation of this distribution. (b) What are the quartiles (the 25% and 75% values) of this distribution? (c) What is the value of the cholesterol level that exceeds 90% of the population? (d) An adult is at moderate risk if cholesterol level is more than one but less than two standard deviations above the mean. What percentage of the population is at moderate risk according to this criterion? (e) An adult is thought to be at high risk if his cholesterol level is more than two standard deviations above the mean. What percentage of the population is at high risk? (f) An adult has low risk if cholesterol level is one standard deviation or more below the mean. What percentage of the population is at low risk?

The line width for semiconductor manufacturing is assumed to be normally distributed with a mean of 0.5 micrometer and a standard deviation of 0.05 micrometer. (a) What is the probability that a line width is greater than 0.62 micrometer? (b) What is the probability that a line width is between 0.47 and 0.63 micrometer? (c) The line width of 90% of samples is below what value?

The fill volume of an automated filling machine used for filling cans of carbonated beverage is normally distributed with a mean of 12.4 fluid ounces and a standard deviation of 0.1 fluid ounce. (a) What is the probability that a fill volume is less than 12 fluid ounces? (b) If all cans less than 12.1 or greater than 12.6 ounces are scrapped, what proportion of cans is scrapped? (c) Determine specifications that are symmetric about the mean that include 99% of all cans

In the previous exercise, suppose that the mean of the filling operation can be adjusted easily, but the standard deviation remains at 0.1 ounce. (a) At what value should the mean be set so that 99.9% of all cans exceed 12 ounces? (b) At what value should the mean be set so that 99.9% of all cans exceed 12 ounces if the standard deviation can be reduced to 0.05 fluid ounce?

The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.4 seconds and a standard deviation of 0.05 seconds. (a) What is the probability that a reaction requires more than 0.5 seconds? (b) What is the probability that a reaction requires between 0.4 and 0.5 seconds? (c) What is the reaction time that is exceeded 90% of the time?

The speed of a file transfer from a server on campus to a personal computer at a students home on a weekday evening is normally distributed with a mean of 60 kilobits per second and a standard deviation of 4 kilobits per second.(a) What is the probability that the file will transfer at a speed of 70 kilobits per second or more? (b) What is the probability that the file will transfer at a speed of less than 58 kilobits per second? (c) If the file is 1 megabyte, what is the average time it will take to transfer the file? (Assume eight bits per byte.)

The average height of a woman aged 2074 years is 64 inches in 2002, with an increase of approximately one inch from 1960 (http://usgovinfo.about.com/od/healthcare). Suppose the height of a woman is normally distributed with a standard deviation of 2 inches. (a) What is the probability that a randomly selected woman in this population is between 58 inches and 70 inches? (b) What are the quartiles of this distribution? (c) Determine the height that is symmetric about the mean that includes 90% of this population. (d) What is the probability that five women selected at random from this population all exceed 68 inches?

In an accelerator center, an experiment needs a 1.41-cmthick aluminum cylinder (http://puhep1.princeton.edu/mumu/ target/Solenoid_Coil.pdf ). Suppose that the thickness of a cylinder has a normal distribution with a mean of 1.41 cm and a standard deviation of 0.01 cm. (a) What is the probability that a thickness is greater than 1.42 cm? (b) What thickness is exceeded by 95% of the samples? (c) If the specifications require that the thickness is between 1.39 cm and 1.43 cm, what proportion of the samples meet specifications?

The demand for water use in Phoenix in 2003 hit a high of about 442 million gallons per day on June 27, 2003 (http://phoenix.gov/WATER/wtrfacts.html). Water use in the summer is normally distributed with a mean of 310 million gallons per day and a standard deviation of 45 million gallons per day. City reservoirs have a combined storage capacity of nearly 350 million gallons. (a) What is the probability that a day requires more water than is stored in city reservoirs? (b) What reservoir capacity is needed so that the probability that it is exceeded is 1%? (c) What amount of water use is exceeded with 95% probability? (d) Water is provided to approximately 1.4 million people. What is the mean daily consumption per person at which the probability that the demand exceeds the current reservoir capacity is 1%? Assume that the standard deviation of demand remains the same

The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours. (a) What is the probability that a laser fails before 5000 hours? (b) What is the life in hours that 95% of the lasers exceed? (c) If three lasers are used in a product and they are assumed to fail independently, what is the probability that all three are still operating after 7000 hours?

The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch and a standard deviation of 0.0004 inch. (a) What is the probability that the diameter of a dot exceeds 0.0026 inch? (b) What is the probability that a diameter is between 0.0014 and 0.0026 inch? (c) What standard deviation of diameters is needed so that the probability in part (b) is 0.995?

The weight of a sophisticated running shoe is normally distributed with a mean of 12 ounces and a standard deviation of 0.5 ounce. (a) What is the probability that a shoe weighs more than 13 ounces? (b) What must the standard deviation of weight be in order for the company to state that 99.9% of its shoes are less than 13 ounces? (c) If the standard deviation remains at 0.5 ounce, what must the mean weight be in order for the company to state that 99.9% of its shoes are less than 13 ounces?

Measurement error that is normally distributed with a mean of zero and a standard deviation of 0.5 gram is added to the true weight of a sample. Then the measurement is rounded to the nearest gram. Suppose that the true weight of a sample is 165.5 grams. (a) What is the probability that the rounded result is 167 grams? (b) What is the probability that the rounded result is 167 grams or greater?

Assume that a random variable is normally distributed with a mean of 24 and a standard deviation of 2. Consider an interval of length one unit that starts at the value a so that the interval is [a, a 1]. For what value of a is the probability of the interval greatest? Does the standard deviation affect that choice of interval?

A study by Bechtel, et al., 2009, in the Archives of Environmental & Occupational Health considered polycyclic aromatic hydrocarbons and immune system function in beef cattle. Some cattle were near major oil- and gas-producing areas of western Canada. The mean monthly exposure to PM1.0 (particulate matter that is  1 m in diameter) was approximately 7.1 g/m3 with standard deviation 1.5. Assume the monthly exposure is normally distributed. (a) What is the probability of a monthly exposure greater than 9 g/m3 ? (b) What is the probability of a monthly exposure between 3 and 8 g/m3 ? (c) What is the monthly exposure level that is exceeded with probability 0.05? (d) What value of mean monthly exposure is needed so that the probability of a monthly exposure greater than 9 g/m3 is 0.01?

An article under review for Air Quality, Atmosphere & Health titled Linking Particulate Matter (PM10) and Childhood Asthma in Central Phoenix used PM10 (particulate matter  10 m in diameter) air quality data measured hourly from sensors in Phoenix, Arizona. The 24-hour (daily) mean PM10 for a centrally located sensor was 50.9 g/m3 with a standard deviation of 25.0. Assume that the daily mean of PM10 is normally distributed. (a) What is the probability of a daily mean of PM10 greater than 100 g/m3 ? (b) What is the probability of a daily mean of PM10 less than 25 g/m3 ? (c) What daily mean of PM10 value is exceeded with probability 5%?

The length of stay at a specific emergency department in Phoenix, Arizona, in 2009 had a mean of 4.6 hours with a standard deviation of 2.9. Assume that the length of stay is normally distributed. (a) What is the probability of a length of stay greater than 10 hours? (b) What length of stay is exceeded by 25% of the visits? (c) From the normally distributed model, what is the probability of a length of stay less than zero hours? Comment on the normally distributed assumption in this example.

Suppose that X is a binomial random variable with and (a) Approximate the probability that X is less than or equal to 70. (b) Approximate the probability that X is greater than 70 and less than 90. (c) Approximate the probability that X 80.

Suppose that X is a Poisson random variable with  6. (a) Compute the exact probability that X is less than 4. (b) Approximate the probability that X is less than 4 and compare to the result in part (a). (c) Approximate the probability that .

Suppose that X has a Poisson distribution with a mean of 64. Approximate the following probabilities: (a) (b) (c)

The manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips. (a) Approximate the probability that more than 25 chips are defective. (b) Approximate the probability that between 20 and 30 chips are defective.

There were 49.7 million people with some type of long-lasting condition or disability living in the United States in 2000. This represented 19.3 percent of the majority of civilians aged five and over (http://factfinder. census.gov). A sample of 1000 persons is selected at random. (a) Approximate the probability that more than 200 persons in the sample have a disability. (b) Approximate the probability that between 180 and 300 people in the sample have a disability.

Phoenix water is provided to approximately 1.4 million people, who are served through more than 362,000 accounts (http://phoenix.gov/WATER/wtrfacts.html). All accounts are metered and billed monthly. The probability that an account has an error in a month is 0.001, and accounts can be assumed to be independent. (a) What is the mean and standard deviation of the number of account errors each month? (b) Approximate the probability of fewer than 350 errors in a month. (c) Approximate a value so that the probability that the number of errors exceeds this value is 0.05. (d) Approximate the probability of more than 400 errors per month in the next two months. Assume that results between months are independent.

An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume that the components fail independently. Approximate the probability that 10 or more of the original 5000 components fail during the useful life of the product. 4-

A corporate Web site contains errors on 50 of 1000 pages. If 100 pages are sampled randomly, without replacement, approximate the probability that at least 1 of the pages in error is in the sample.

Suppose that the number of asbestos particles in a sample of 1 squared centimeter of dust is a Poisson random variable with a mean of 1000. What is the probability that 10 squared centimeters of dust contains more than 10,000 particles?

A high-volume printer produces minor print-quality errors on a test pattern of 1000 pages of text according to a Poisson distribution with a mean of 0.4 per page. (a) Why are the numbers of errors on each page independent random variables? (b) What is the mean number of pages with errors (one or more)? (c) Approximate the probability that more than 350 pages contain errors (one or more).

Hits to a high-volume Web site are assumed to follow a Poisson distribution with a mean of 10,000 per day. Approximate each of the following: (a) The probability of more than 20,000 hits in a day (b) The probability of less than 9900 hits in a day (c) The value such that the probability that the number of hits in a day exceeds the value is 0.01 (d) Approximate the expected number of days in a year (365 days) that exceed 10,200 hits. (e) Approximate the probability that over a year (365 days) more than 15 days each have more than 10,200 hits.

An acticle in Biometrics [Integrative Analysis of Transcriptomic and Proteomic Data of Desulfovibrio vulgaris: A Nonlinear Model to Predict Abundance of Undetected Proteins (2009)] found that protein abundance from an operon (a set of biologically related genes) was less dispersed than from randomly selected genes. In the research, 1000 sets of genes were randomly constructed and 75% of these sets were more disperse than a specific opteron. If the probability that a random set is more disperse than this opteron is truly 0.5, approximate the probability that 750 or more random sets exceed the opteron. From this result, what do you conclude about the dispersion in the opteron versus random genes?

An article under review for Air Quality, Atmosphere & Health titled Linking Particulate Matter (PM10) and Childhood Asthma in Central Phoenix linked air quality to childhood asthma incidents. The study region in central Phoenix, Arizona, recorded 10,500 asthma incidents in children in a 21-month period. Assume that the number of asthma incidents follows a Poisson distribution.(a) Approximate the probability of more than 550 asthma incidents in a month. (b) Approximate the probability of 450 to 550 asthma incidents in a month. (c) Approximate the number of asthma incidents exceeded with probability 5%. (d) If the number of asthma incidents were greater during the winter than the summer, what would this imply about the Poisson distribution assumption?.

Suppose X has an exponential distribution with  2. Determine the following: (a) (b) (c) (d) (e) Find the value of x such that

Suppose X has an exponential distribution with mean equal to 10. Determine the following: (a) (b) (c) (d) Find the value of x such that

Suppose X has an exponential distribution with a mean of 10. Determine the following: (a) (b) (c) Compare the results in parts (a) and (b) and comment on the role of the lack of memory property.

Suppose the counts recorded by a Geiger counter follow a Poisson process with an average of two counts per minute. (a) What is the probability that there are no counts in a 30-second interval? (b) What is the probability that the first count occurs in less than 10 seconds? (c) What is the probability that the first count occurs between 1 and 2 minutes after start-up?

Suppose that the log-ons to a computer network follow a Poisson process with an average of 3 counts per minute. (a) What is the mean time between counts? (b) What is the standard deviation of the time between counts? (c) Determine x such that the probability that at least one count occurs before time x minutes is 0.95.

The time between calls to a plumbing supply business is exponentially distributed with a mean time between calls of 15 minutes. (a) What is the probability that there are no calls within a 30-minute interval? (b) What is the probability that at least one call arrives within a 10-minute interval? (c) What is the probability that the first call arrives within 5 and 10 minutes after opening? (d) Determine the length of an interval of time such that the probability of at least one call in the interval is 0.90. 4-9

The life of automobile voltage regulators has an exponential distribution with a mean life of six years. You purchase an automobile that is six years old, with a working voltage regulator, and plan to own it for six years. (a) What is the probability that the voltage regulator fails during your ownership? (b) If your regulator fails after you own the automobile three years and it is replaced, what is the mean time until the next failure?

Suppose that the time to failure (in hours) of fans in a personal computer can be modeled by an exponential distribution with (a) What proportion of the fans will last at least 10,000 hours? (b) What proportion of the fans will last at most 7000 hours?

The time between the arrival of electronic messages at your computer is exponentially distributed with a mean of two hours. (a) What is the probability that you do not receive a message during a two-hour period? (b) If you have not had a message in the last four hours, what is the probability that you do not receive a message in the next two hours? (c) What is the expected time between your fifth and sixth messages?

The time between arrivals of taxis at a busy intersection is exponentially distributed with a mean of 10 minutes.(a) What is the probability that you wait longer than one hour for a taxi? (b) Suppose you have already been waiting for one hour for a taxi. What is the probability that one arrives within the next 10 minutes? (c) Determine x such that the probability that you wait more than x minutes is 0.10. (d) Determine x such that the probability that you wait less than x minutes is 0.90. (e) Determine x such that the probability that you wait less than x minutes is 0.50.

The number of stork sightings on a route in South Carolina follows a Poisson process with a mean of 2.3 per year. (a) What is the mean time between sightings? (b) What is the probability that there are no sightings within three months (0.25 years)? (c) What is the probability that the time until the first sighting exceeds six months? (d) What is the probability of no sighting within three years?

According to results from the analysis of chocolate bars in Chapter 3, the mean number of insect fragments was 14.4 in 225 grams. Assume the number of fragments follows a Poisson distribution. (a) What is the mean number of grams of chocolate until a fragment is detected? (b) What is the probability that there are no fragments in a 28.35-gram (one-ounce) chocolate bar? (c) Suppose you consume seven one-ounce (28.35-gram) bars this week. What is the probability of no insect fragments?

The distance between major cracks in a highway follows an exponential distribution with a mean of 5 miles. (a) What is the probability that there are no major cracks in a 10-mile stretch of the highway? (b) What is the probability that there are two major cracks in a 10-mile stretch of the highway? (c) What is the standard deviation of the distance between major cracks? (d) What is the probability that the first major crack occurs between 12 and 15 miles of the start of inspection? (e) What is the probability that there are no major cracks in two separate 5-mile stretches of the highway? (f ) Given that there are no cracks in the first 5 miles inspected, what is the probability that there are no major cracks in the next 10 miles inspected?

The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours. (a) What is the probability that an assembly on test fails in less than 100 hours? (b) What is the probability that an assembly operates for more than 500 hours before failure? (c) If an assembly has been on test for 400 hours without a failure, what is the probability of a failure in the next 100 hours?(d) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently. (e) If 10 assemblies are tested, what is the probability that all have failed by 800 hours? Assume the assemblies fail independently.(d) If 10 assemblies are tested, what is the probability that at least one fails in less than 100 hours? Assume that the assemblies fail independently. (e) If 10 assemblies are tested, what is the probability that all have failed by 800 hours? Assume the assemblies fail independently.

The time between arrivals of small aircraft at a county airport is exponentially distributed with a mean of one hour. (a) What is the probability that more than three aircraft arrive within an hour? (b) If 30 separate one-hour intervals are chosen, what is the probability that no interval contains more than three arrivals? (c) Determine the length of an interval of time (in hours) such that the probability that no arrivals occur during the interval is 0.10.

The time between calls to a corporate office is exponentially distributed with a mean of 10 minutes. (a) What is the probability that there are more than three calls in one-half hour? (b) What is the probability that there are no calls within onehalf hour? (c) Determine x such that the probability that there are no calls within x hours is 0.01. (d) What is the probability that there are no calls within a twohour interval? (e) If four nonoverlapping one-half-hour intervals are selected, what is the probability that none of these intervals contains any call? (f ) Explain the relationship between the results in part (a) and (b).

Assume that the flaws along a magnetic tape follow a Poisson distribution with a mean of 0.2 flaw per meter. Let X denote the distance between two successive flaws. (a) What is the mean of X? (b) What is the probability that there are no flaws in 10 consecutive meters of tape? (c) Does your answer to part (b) change if the 10 meters are not consecutive? (d) How many meters of tape need to be inspected so that the probability that at least one flaw is found is 90%? (e) What is the probability that the first time the distance between two flaws exceeds 8 meters is at the fifth flaw? (f ) What is the mean number of flaws before a distance between two flaws exceeds 8 meters?

If the random variable X has an exponential distribution with mean , determine the following: (a) (b) (c) (d) How do the results depend on ?

Derive the formula for the mean and variance of an exponential random variable.

Web crawlers need to estimate the frequency of changes to Web sites to maintain a current index for Web searches. Assume that the changes to a Web site follow a Poisson process with a mean of 3.5 days. (a) What is the probability that the next change occurs in less than two days? (b) What is the probability that the next change occurs in greater than seven days? (c) What is the time of the next change that is exceeded with probability 90%? (d) What is the probability that the next change occurs in less than 10 days, given that it has not yet occurred after three days?

The length of stay at a specific emergency department in Phoenix, Arizona, had a mean of 4.6 hours. Assume that the length of stay is exponentially distributed. (a) What is the standard deviation of the length of stay? (b) What is the probability of a length of stay greater than 10 hours? (c) What length of stay is exceeded by 25% of the visits?

Use the properties of the gamma function to evaluate the following: (a) (b) (c)

Given the probability density function determine the mean and variance of the distribution.

Calls to a telephone system follow a Poisson distribution with a mean of five calls per minute. (a) What is the name applied to the distribution and parameter values of the time until the tenth call? (b) What is the mean time until the tenth call? (c) What is the mean time between the ninth and tenth calls?(d) What is the probability that exactly four calls occur within one minute? (e) If 10 separate 1-minute intervals are chosen, what is the probability that all intervals contain more than two calls?

Raw materials are studied for contamination. Suppose that the number of particles of contamination per pound of material is a Poisson random variable with a mean of 0.01 particle per pound. (a) What is the expected number of pounds of material required to obtain 15 particles of contamination? (b) What is the standard deviation of the pounds of materials required to obtain 15 particles of contamination?

The time between failures of a laser in a cytogenics machine is exponentially distributed with a mean of 25,000 hours. (a) What is the expected time until the second failure? (b) What is the probability that the time until the third failure exceeds 50,000 hours?

In a data communication system, several messages that arrive at a node are bundled into a packet before they are transmitted over the network. Assume the messages arrive at the node according to a Poisson process with messages per minute. Five messages are used to form a packet. (a) What is the mean time until a packet is formed, that is, until five messages have arrived at the node? (b) What is the standard deviation of the time until a packet is formed? (c) What is the probability that a packet is formed in less than 10 seconds? (d) What is the probability that a packet is formed in less than 5 seconds?

Errors caused by contamination on optical disks occur at the rate of one error every bits. Assume the errors follow a Poisson distribution. (a) What is the mean number of bits until five errors occur? (b) What is the standard deviation of the number of bits until five errors occur?(c) The error-correcting code might be ineffective if there are three or more errors within bits. What is the probability of this event?

Calls to the help line of a large computer distributor follow a Poisson distribution with a mean of 20 calls per minute. (a) What is the mean time until the one-hundredth call? (b) What is the mean time between call numbers 50 and 80? (c) What is the probability that three or more calls occur within 15 seconds?

The time between arrivals of customers at an automatic teller machine is an exponential random variable with a mean of 5 minutes. (a) What is the probability that more than three customers arrive in 10 minutes? (b) What is the probability that the time until the fifth customer arrives is less than 15 minutes?

Use integration by parts to show that

Show that the gamma density function integrates to 1.

Use the result for the gamma distribution to determine the mean and variance of a chi-square distribution with r 72.

Patients arrive at an emergency department according to a Poisson process with a mean of 6.5 per hour. (a) What is the mean time until the tenth arrival? (b) What is the probability that more than 20 minutes is required for the third arrival?

The total service time of a multistep manufacturing operation has a gamma distribution with mean 18 minutes and standard deviation 6. (a) Determine the parameters  and r of the distribution. (b) Assume each step has the same distribution for service time. What distribution for each step and how many steps produce this gamma distribution of total service time?

Suppose that X has a Weibull distribution with and hours. Determine the mean and variance of X.

Suppose that X has a Weibull distribution with and hours. Determine the following: (a) (b) 4-127. If X is a Weibull random variable with  1 and  1000, what is another name for the distribution of X and what is the mean of X?

If X is a Weibull random variable with  1 and  1000, what is another name for the distribution of X and what is the mean of X?

Assume that the life of a roller bearing follows a Weibull distribution with parameters and hours. (a) Determine the probability that a bearing lasts at least 8000 hours. (b) Determine the mean time until failure of a bearing. (c) If 10 bearings are in use and failures occur independently, what is the pr

The life (in hours) of a computer processing unit (CPU) is modeled by a Weibull distribution with parameters and hours. (a) Determine the mean life of the CPU. (b) Determine the variance of the life of the CPU. (c) What is the probability that the CPU fails before 500 hours?

Assume the life of a packaged magnetic disk exposed to corrosive gases has a Weibull distribution with and the mean life is 600 hours. (a) Determine the probability that a packaged disk lasts at least 500 hours. (b) Determine the probability that a packaged disk fails before 400 hours.

The life (in hours) of a magnetic resonance imaging machine (MRI) is modeled by a Weibull distribution with parameters and hours. (a) Determine the mean life of the MRI. (b) Determine the variance of the life of the MRI. (c) What is the probability that the MRI fails before 250 hours?

An article in the Journal of the Indian Geophysical Union titled Weibull and Gamma Distributions for Wave. Parameter Predictions (2005, Vol. 9, pp. 5564) used the Weibull distribution to model ocean wave heights. Assume that the mean wave height at the observation station is 2.5 m and the shape parameter equals 2. Determine the standard deviation of wave height.

An article in the Journal of Geophysical Research [Spatial and Temporal Distributions of U.S. of Winds and Wind Power at 80 m Derived from Measurements, (2003, vol. 108, pp. 101: 1020)] considered wind speed at stations throughout the U.S. A Weibull distribution can be used to model the distribution of wind speeds at a given location. Every location is characterized by a particular shape and scale parameter. For a station at Amarillo, Texas, the mean wind speed at 80 m (the hub height of large wind turbines) in 2000 was 10.3 m/s with a standard deviation of 4.9 m/s. Determine the shape and scale parameters of a Weibull distribution with these properties.

Suppose that X has a Weibull distribution with  2 and  8.6. Determine the following: (a) P(X  10) (b) P(X  9) (c) P(8  X  11) (d) value for x such that P(X  x) 0.9 4-

Suppose the lifetime of a component (in hours) is modeled with a Weibull distribution with  2 and  4000. Determine the following: (a) P(X  3000) (b) P(X  6000 | X  3000) (c) Comment on the probabilities in the previous parts compared to the results for an exponential distribution. 4

Suppose the lifetime of a component (in hours) is modeled with a Weibull distribution with  0.5 and  = 4000. Determine the following: (a) P(X  3000) (b) P(X  6000 | X  3000) (c) Comment on the probabilities in the previous parts compared to the results for an exponential distribution. (d) Comment on the role of the parameter  in a lifetime model with the Weibull distribution.

Suppose X has a Weibull distribution with  2 and  2000.(a) Determine P(X 3500). (b) Determine P(X 3500) for an exponential random variable with the same mean as the Weibull distribution. (c) Suppose X represent the lifetime of a component in hours. Comment on the probability that the lifetime exceeds 3500 hours under the Weibull and exponential distributions.

Suppose that X has a lognormal distribution with parameters and . Determine the following: (a) (b) The value for x such that (c) The mean and variance of X

Suppose that X has a lognormal distribution with parameters and . Determine the following: (a) (b) The value for x such that (c) The mean and variance of X

Suppose that X has a lognormal distribution with parameters and . Determine the following: (a) (b) The conditional probability that given that (c) What does the difference between the probabilities in parts (a) and (b) imply about lifetimes of lognormal random variables?

The length of time (in seconds) that a user views a page on a Web site before moving to another page is a lognormal random variable with parameters and . (a) What is the probability that a page is viewed for more than 10 seconds?(b) By what length of time have 50% of the users moved to another page? (c) What is the mean and standard deviation of the time until a user moves from the page?

Suppose that X has a lognormal distribution and that the mean and variance of X are 100 and 85,000, respectively. Determine the parameters and of the lognormal distribution. (Hint: define and and write two equations in terms of x and y.)

The lifetime of a semiconductor laser has a lognormal distribution, and it is known that the mean and standard deviation of lifetime are 10,000 and 20,000, respectively. (a) Calculate the parameters of the lognormal distribution. (b) Determine the probability that a lifetime exceeds 10,000 hours. (c) Determine the lifetime that is exceeded by 90% of lasers

An article in Health and Population: Perspectives and Issues(2000, Vol. 23, pp. 2836) used the lognormal distribution to model blood pressure in humans. The mean systolic blood y exp12 x exp12 2 2  pressure (SBP) in males age 17 was 120.87 mm Hg. If the coefficient of variation (100%  standard deviation/mean) is 9%, what are the parameter values of the lognormal distribution?

Derive the probability density function of a lognormal random variable from the derivative of the cumulative distribution function.

Suppose X has a lognormal distribution with parameters  10 and 2 16. Determine the following: (a) (b) (c) value exceeded with probability 0.7 4

Suppose the length of stay (in hours) at an emergency department is modeled with a lognormal random variable X with  1.5 and  0.4. Determine the following: (a) mean and variance (b) (c) Comment on the difference between the probability calculated from this lognormal distribution and a normal distribution with the same mean and variance. P

Suppose X has a beta distribution with parameters  2.5 and  2.5. Sketch an approximate graph of the probability density function. Is the density symmetric?

Suppose x has a beta distribution with parameters  2.5 and  1. Determine the following: (a) P(X  0.25) (b) P(0.25  X  0.75) (c) mean and variance 4

Suppose X has a beta distribution with parameters  = 1 and  = 4.2. Determine the following: (a) P(X  0.25) (b) P(0.5  X ) (c) mean and variance

A European standard value for a low-emission window glazing uses 0.59 as the proportion of solar energy that enters a room. Suppose that the distribution of the proportion of solar energy that enters a room is a beta random variable. (a) Calculate the mode, mean, and variance of the distribution for  3 and  1.4. (b) Calculate the mode, mean, and variance of the distribution for  10 and  6.25. (c) Comment on the difference in dispersion in the distributions from the previous parts. 4-15

The length of stay at an emergency department is the sum of the waiting and service times. Let X denote the proportion of time spent waiting and assume a beta distribution with  10 and  1. Determine the following: (a) P(X  0.9) (b) P(X  0.5) (c) mean and variance 4

The maximum time to complete a task in a project is 2.5 days. Suppose that the completion time as a proportion of this maximum is a beta random variable with  2 and  3. What is the probability that the task requires more than two days to complete? S

The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample is seconds. (a) What percentage of tests require more than 70 seconds to complete? (b) What percentage of tests require less than one minute to complete? (c) Determine the mean and variance of the time to complete a test on a sample.

The tensile strength of paper is modeled by a normal distribution with a mean of 35 pounds per square inch and a standard deviation of 2 pounds per square inch. (a) What is the probability that the strength of a sample is less than 40 lb/in2 ?(b) If the specifications require the tensile strength to exceed 30 lb/in2 , what proportion of the samples is scrapped?

The time it takes a cell to divide (called mitosis) is normally distributed with an average time of one hour and a standard deviation of 5 minutes. (a) What is the probability that a cell divides in less than 45 minutes? (b) What is the probability that it takes a cell more than 65 minutes to divide? (c) By what time have approximately 99% of all cells completed mitosis?

The length of an injection-molded plastic case that holds magnetic tape is normally distributed with a length of 90.2 millimeters and a standard deviation of 0.1 millimeter. (a) What is the probability that a part is longer than 90.3 millimeters or shorter than 89.7 millimeters? (b) What should the process mean be set at to obtain the greatest number of parts between 89.7 and 90.3 millimeters? (c) If parts that are not between 89.7 and 90.3 millimeters are scrapped, what is the yield for the process mean that you selected in part (b)? Assume that the process is centered so that the mean is 90 millimeters and the standard deviation is 0.1 millimeter. Suppose that 10 cases are measured, and they are assumed to be independent. (d) What is the probability that all 10 cases are between 89.7 and 90.3 millimeters? (e) What is the expected number of the 10 cases that are between 89.7 and 90.3 millimeters?

The sick-leave time of employees in a firm in a month is normally distributed with a mean of 100 hours and a standard deviation of 20 hours. (a) What is the probability that the sick-leave time for next month will be between 50 and 80 hours? (b) How much time should be budgeted for sick leave if the budgeted amount should be exceeded with a probability of only 10%?

The percentage of people exposed to a bacteria who become ill is 20%. Assume that people are independent. Assume that 1000 people are exposed to the bacteria. Approximate each of the following: (a) The probability that more than 225 become ill (b) The probability that between 175 and 225 become ill (c) The value such that the probability that the number of people who become ill exceeds the value is 0.01

The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with (a) What is the probability that the laser will last at least 20,000 hours? (b) What is the probability that the laser will last at most 30,000 hours? (c) What is the probability that the laser will last between 20,000 and 30,000 hours?

When a bus service reduces fares, a particular trip from New York City to Albany, New York, is very popular. A small bus can carry four passengers. The time between calls for tickets is exponentially distributed with a mean of 30 minutes. Assume that each call orders one ticket. What is the probability that the bus is filled in less than three hours from the time of the fare reduction?

The time between process problems in a manufacturing line is exponentially distributed with a mean of 30 days. (a) What is the expected time until the fourth problem? (b) What is the probability that the time until the fourth problem exceeds 120 days?

The life of a recirculating pump follows a Weibull distribution with parameters and hours. (a) Determine the mean life of a pump. (b) Determine the variance of the life of a pump. (c) What is the probability that a pump will last longer than its mean?

The size of silver particles in a photographic emulsion is known to have a log normal distribution with a mean of 0.001 mm and a standard deviation of 0.002 mm. (a) Determine the parameter values for the lognormal distribution. (b) What is the probability of a particle size greater than 0.005 mm?

Suppose that for Determine the following: (a) (b) (c) (d) Determine the cumulative distribution function of the random variable. (e) Determine the mean and variance of the random variable.

The time between calls is exponentially distributed with a mean time between calls of 10 minutes. (a) What is the probability that the time until the first call is less than 5 minutes? (b) What is the probability that the time until the first call is between 5 and 15 minutes? (c) Determine the length of an interval of time such that the probability of at least one call in the interval is 0.90. (d) If there has not been a call in 10 minutes, what is the probability that the time until the next call is less than 5 minutes? (e) What is the probability that there are no calls in the intervals from 10:00 to 10:05, from 11:30 to 11:35, and from 2:00 to 2:05? (f) What is the probability that the time until the third call is greater than 30 minutes? (g) What is the mean time until the fifth call?

The CPU of a personal computer has a lifetime that is exponentially distributed with a mean lifetime of six years. You have owned this CPU for three years.(a) What is the probability that the CPU fails in the next three years? (b) Assume that your corporation has owned 10 CPUs for three years, and assume that the CPUs fail independently. What is the probability that at least one fails within the next three years?

Suppose that X has a lognormal distribution with parameters and . Determine the following: (a) (b) The value for x such that (c) The mean and variance of X

Suppose that X has a lognormal distribution and that the mean and variance of X are 50 and 4000, respectively. Determine the following: (a) The parameters and of the lognormal distribution (b) The probability that X is less than 150

Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per square centimeter of surface dust is 100. A sample of 800 square centimeters of dust is analyzed. Assume a particular grid cell under the microscope represents 1/160,000 of the sample. (a) What is the probability that at least one fiber is visible in the grid cell? (b) What is the mean of the number of grid cells that need to be viewed to observe 10 that contain fibers? (c) What is the standard deviation of the number of grid cells that need to be viewed to observe 10 that contain fibers?

Without an automated irrigation system, the height of plants two weeks after germination is normally distributed with a mean of 2.5 centimeters and a standard deviation of 0.5 centimeter. (a) What is the probability that a plants height is greater than 2.25 centimeters? (b) What is the probability that a plants height is between 2.0 and 3.0 centimeters? (c) What height is exceeded by 90% of the plants?

Continuation of Exercise 4-171. With an automated irrigation system, a plant grows to a height of 3.5 centimeters two weeks after germination. (a) What is the probability of obtaining a plant of this height or greater from the distribution of heights in Exercise 4-165. (b) Do you think the automated irrigation system increases the plant height at two weeks after germination?

The thickness of a laminated covering for a wood surface is normally distributed with a mean of 5 millimeters and a standard deviation of 0.2 millimeter. (a) What is the probability that a covering thickness is greater than 5.5 millimeters? (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?

The diameter of the dot produced by a printer is normally 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?

Continuation of Exercise 4-174. Assume that the standard 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 specifications are to be chosen symmetrically around the mean of 0.002.

The life of a semiconductor laser at a constant power is normally distributed with a mean of 7000 hours and a standard deviation of 600 hours. (a) What is the probability that a laser fails before 5800 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 independently. What should the mean life equal in order for 99% of the products to exceed 10,000 hours before failure?

Continuation of Exercise 4-176. Rework parts (a) and (b). Assume that the lifetime is an exponential random variable with the same mean.

Continuation of Exercise 4-176. Rework parts (a) and (b). Assume that the lifetime is a lognormal random variable with the same mean and standard deviation.

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

An airline makes 200 reservations for a flight who holds 185 passengers. The probability that a passenger arrives for the flight is 0.9 and the passengers are assumed to be independent. (a) Approximate the probability that all the passengers who 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. [Hint: Successively try values for the number of reservations.]

The steps in this exercise lead to the probability density function of an Erlang random variable X with parameters and and (a) Use the Poisson distribution to express . (b) Use the result from part (a) to determine the cumulative distribution function of X. (c) Differentiate the cumulative distribution function in part (b) and simplify to obtain the probability density function of X.

A bearing assembly contains 10 bearings. The bearing diameters are assumed to be independent and normally distributed with a mean of 1.5 millimeters and a standard deviation of 0.025 millimeter. What is the probability that the maximum diameter bearing in the assembly exceeds 1.6 millimeters?

Let the random variable X denote a measurement from a manufactured product. Suppose the target value for the measurement is m. For example, X could denote a dimensional length, and the target might be 10 millimeters. The quality loss of the process producing the product is defined to be the expected value of , where k is a constant that relates a deviation from target to a loss measured in dollars. (a) Suppose X is a continuous random variable with and . What is the quality loss of the process? (b) Suppose X is a continuous random variable with and . What is the quality loss of the process?

The lifetime of an electronic amplifier is modeled as an exponential random variable. If 10% of the amplifiers have a mean of 20,000 hours and the remaining amplifiers have a mean of 50,000 hours, what proportion of the amplifiers fail before 60,000 hours? 4-185. Lack of Memory Property. Show that for an exponential random

A process is said to be of six-sigma quality if the process mean is at least six standard deviations from the nearest specification. Assume a normally distributed measurement.(a) If a process mean is centered between upper and lower specifications at a distance of six standard deviations from each, what is the probability that a product does not meet specifications? Using the result that 0.000001 equals one part per million, express the answer in parts per million. (b) Because it is difficult to maintain a process mean centered between the specifications, the probability of a product not meeting specifications is often calculated after assuming the process shifts. If the process mean positioned as in part (a) shifts upward by 1.5 standard deviations, what is the probability that a product does not meet specifications? Express the answer in parts per million. (c) Rework part (a). Assume that the process mean is at a distance of three standard deviations. (d) Rework part (b). Assume that the process mean is at a distance of three standard deviations and then shifts upward by 1.5 standard deviations. (e) Compare the results in parts (b) and (d) and comment.