Chapter 4

The diameter of a particle of contamination (in micrometers) is modeled with the probability density function \(f(x)=2 / x^{3}\) for \(x>1 .\) Determine the following: (a) \(P(X<2)\) (b) \(P(X>5)\) (c) \(P(48)\) (e) \(x\) such that \(P(X

Suppose that \(f(x)=1.5 x^{2}\) for \(-1-0.5)\) (f) \(x\) such that \(P(x

The probability density function of the time to failure of an electronic component in a copier (in hours) is \(f(x)=\) \(e^{-x / 1000 / 1000}\) for \(x>0 .\) 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) The number of hours at which \(10 \%\) of all components have failed

The probability density function of the length of a cutting blade is \(f(x)=1.25\) for \(74.675.2)\) (c) If the specifications for this process are from 74.7 to 75.3

The probability density function of the length of a metal rod is \(f(x)=2\) for \(2.3

An article in Electric Power Systems Research ["Modeling Real-Time Balancing Power Demands in Wind Power Systems Using Stochastic Differential Equations" (2010, Vol. \(80(8),\) pp. \(966-974\) ) ] considered a new probabilistic model to balance power demand with large amounts of wind power. In this model, the power loss from shutdowns is assumed to have a triangular distribution with probability density function $$ f(x)=\left\\{\begin{array}{cc} -5.56 \times 10^{-4}+5.56 \times 10^{-6} x, & x \in[100,500] \\ 4.44 \times 10^{-3}-4.44 \times 10^{-6} x, & x \in[500,1000] \\ 0, & \text { otherwise } \end{array}\right. $$ Determine the following: (a) \(P(X<90)\) (b) \(P(100800)\) (d) Value exceeded with probability 0.1 .

A test instrument needs to be calibrated periodically to prevent measurement errors. After some time of use without calibration, it is known that the probability density function of the measurement error is \(f(x)=1-0.5 x\) for \(0

The distribution of \(X\) is approximated with a triangular probability density function \(f(x)=0.025 x-0.0375\) for \(30

The waiting time for service at a hospital emergency department (in hours) follows a distribution with probability density function \(f(x)=0.5 \exp (-0.5 x)\) for \(02)\) (c) Value \(x\) (in hours) exceeded with probability 0.05 .

If \(X\) is a continuous random variable, argue that \(P\left(x_{1} \leq X \leq x_{2}\right)=P\left(x_{1}

Suppose that the cumulative distribution function of the random variable \(X\) is $$ F(x)=\left\\{\begin{array}{lr} 0 & x<0 \\ 0.25 x & 0 \leq x<5 \\ 1 & 5 \leq x \end{array}\right. $$ Determine the following: (a) \(P(X<2.8)\) (b) \(P(X>1.5)\) (c) \(P(X<-2)\) (d) \(P(X>6)\)

Suppose that the cumulative distribution function of the random variable \(X\) is \(F(x)=\left\\{\begin{array}{lc}0 & x<-2 \\ 0.25 x+0.5 & -2 \leq x<2 \\ 1 & 2 \leq x\end{array}\right.\) Determine the following: (a) \(P(X<1.8)\) (b) \(P(X>-1.5)\) (c) \(P(X<-2)\) (d) \(P(-1

The probability density function of the time you arrive at a terminal (in minutes after 8: 00 A.m. ) is \(f(x)=0.1 \exp (-0.1 x)\) for \(0

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

$$ F(x)=\left\\{\begin{array}{lr} 0 & x<0 \\ 0.2 x & 0 \leq x<4 \\ 0.04 x+0.64 & 4 \leq x<9 \\ 1 & 9 \leq x \end{array}\right. $$

$$ F(x)=\left\\{\begin{array}{lc} 0 & x<-2 \\ 0.25 x+0.5 & -2 \leq x<1 \\ 0.5 x+0.25 & 1 \leq x<1.5 \\ 1 & 1.5 \leq x \end{array}\right. $$

Suppose that \(f(x)=0.25\) for \(0

Suppose that 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

The thickness of a conductive coating in micrometers has a density function of \(600 x^{-2}\) for \(100 \mu \mathrm{m}

The probability density function of the weight of packages delivered by a post office is \(f(x)=70 /\left(69 x^{2}\right)\) for \(1

Integration by parts is required. The probability density function for the diameter of a drilled hole in millimeters is \(10 e^{-10(x-5)}\) for \(x>5 \mathrm{~mm}\). Although the target diameter is 5 millimeters, vibrations, tool wear, and other nuisances produce diameters greater 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 that \(X\) has a continuous uniform distribution over the interval \([1.5,5.5] .\) Determine the following: (a) Mean, variance, and standard deviation of \(X\) (b) \(P(X<2.5)\). (c) Cumulative distribution function

Suppose \(X\) has a continuous uniform distribution over the interval \([-1,1] .\) Determine the following: (a) Mean, variance, and standard deviation of \(X\) (b) Value for \(x\) such that \(P(-x

The net weight in pounds of a packaged chemical herbicide is uniform for \(49.75

The net weight in pounds of a packaged chemical herbicide is uniform for \(49.75

Suppose that 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 are 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. Determine the following: (a) Cumulative distribution function of photoresist thickness (b) Proportion of wafers that exceeds 0.2125 micrometers in photoresist thickness (c) Thickness exceeded by \(10 \%\) of the wafers (d) 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 a person's 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 the following: (a) Cumulative distribution function of the time (in minutes) between arrival and 8: 30 A.M. (b) Mean and variance of the distribution in the previous part (c) Probability that a visitor waits less than 10 minutes for a show (d) 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 shampoo more than 375 milliliters in the container 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 a circuit's true voltage. 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 are the mean and variance of the measured voltage?

A beacon transmits a signal every 10 minutes (such as \(8: 20,8: 30,\) etc. \()\). The time at which a receiver is tuned to detect the beacon is a continuous uniform distribution from 8: 00 A.M. to 9: 00 A.M. Consider the waiting time until the next signal from the beacon is received. (a) Is it reasonable to model the waiting time as a continuous uniform distribution? Explain. (b) What is the mean waiting time? (c) What is the probability that the waiting time is less than 3 minutes?

An electron emitter produces electron beams with changing kinetic energy that is uniformly distributed between three and seven joules. Suppose that it is possible to adjust the upper limit of the kinetic energy (currently set to seven joules). (a) What is the mean kinetic energy? (b) What is the variance of the kinetic energy? (c) What is the probability that an electron beam has a kinetic energy of exactly 3.2 joules? (d) What should be the upper limit so that the mean kinetic energy increases to eight joules? (e) What should be the upper limit so that the variance of kinetic energy decreases to 0.75 joules?

Assume that \(X\) 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

Assume that \(X\) 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

Assume that \(X\) 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

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 sample's strength is less than \(6250 \mathrm{Kg} / \mathrm{cm}^{2} ?\) (b) What is the probability that a sample's strength is between 5800 and \(5900 \mathrm{Kg} / \mathrm{cm}^{2} ?\) (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. 273-279)] showed a mean time of 129 minutes and a standard deviation of 14 minutes for anterior cruciate ligament (ACL) reconstruction surgery at highvolume 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 high-volume 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 \(120-240 \mathrm{mg} / \mathrm{dl}\). The Food and Nutrition Institute of the Philippines found that the total cholesterol level for Filipino adults has a mean of \(159.2 \mathrm{mg} / \mathrm{dl}\) and \(84.1 \%\) of adults have a cholesterol level less than \(200 \mathrm{mg} / \mathrm{dl}\) (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 \(75 \%\) percentiles) 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 whose cholesterol level is more than two standard deviations above the mean is thought to be at high risk. What percentage of the population is at high risk? (f) An adult whose cholesterol level is less than one standard deviations below the mean is thought to be at low risk. 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 more 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.

A driver's reaction time 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 reaction time is exceeded \(90 \%\) of the time?

The speed of a file transfer from a server on campus to a personal computer at a student's home on a weekday evening is normally distributed with a mean of 60 kilobits per second and a standard deviation of four 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 one megabyte, what is the average time it will take to transfer the file? (Assume eight bits per byte.)

In 2002 ,the average height of a woman aged \(20-74\) years was 64 inches with an increase of approximately 1 inch from 1960 (http://usgovinfo.about.com/od/healthcare). Suppose the height of a woman is normally distributed with a standard deviation of two 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 \(1.41-\mathrm{cm}-\) thick 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 \(1.41 \mathrm{~cm}\) and a standard deviation of \(0.01 \mathrm{~cm} .\) (a) What is the probability that a thickness is greater than \(1.42 \mathrm{~cm}\) ? (b) What thickness is exceeded by \(95 \%\) of the samples? (c) If the specifications require that the thickness is between \(1.39 \mathrm{~cm}\) and \(1.43 \mathrm{~cm}\), what proportion of the samples meets specifications?

Go Tutorial The demand for water use in Phoenix in 2003 hit a high of about 442 million gallons per day on June 27 (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 \(0.0026 ?\) (b) What is the probability that a diameter is between 0.0014 and \(0.0026 ?\) (c) What standard deviation of diameters is needed so that the probability in part (b) is \(0.995 ?\)