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?
> Determine the cumulative distribution function for the distribution in Exercise 4.1.7. Use the cumulative distribution function to determine the probability that a length exceeds 2.7 meters.
> An article in IEEE Journal on Selected Areas in Communications [“Impulse Response Modeling of Indoor Radio Propagation Channels” (1993, Vol. 11(7), pp. 967–978)] indicated that the successful design of indoor communication systems requires characterizati
> Suppose that the construction of a solar power station is initiated. The project’s completion time has not been set due to uncertainties in financial resources. The proportion of completion within one year has a beta distribution with parameters α = 1 an
> 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 independent. a. Approximate the probability that all the passengers who arrive
> A square inch of carpeting contains 50 carpet fibers. The probability of a damaged fiber is 0.0001. Assume that the damaged fibers occur independently. a. Approximate the probability of one or more damaged fibers in one square yard of carpeting. b. Appro
> The diameter of the dot produced by a printer is normally distributed with a mean diameter of 0.002 inch. a. 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
> A random variable X has the gamma distribution a. Show that the moment-generating function for t b. Find the mean and variance of X.
> The waiting time for service at a hospital emergency department follows an exponential distribution with a mean of three hours. Determine the following: a. Waiting time is greater than four hours b. Waiting time is greater than six hours given that you h
> 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
> Suppose that X has a lognormal distribution and that the mean and variance of X are 50 and 4000, respectively. Determine the following: a. Parameters θ and ω2 of the lognormal distribution b. Probability that X is less than 150
> Suppose that X has a lognormal distribution with parameters θ = 0 and ω2 = 4. Determine the following: a. P(10 < X < 50) b. Value for x such that P(X < x) = 0.05 c. Mean and variance of X
> Suppose that f (x) = 0.5x − 1 for 2 < x < 4. Determine the following: a. P(X < 2.5) b. P(X > 3) c. P(2.5 < X < 3.5) d. Determine the cumulative distribution function of therandom 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 five minutes? b. What is the probability that the time until the first call is be
> The life of a recirculating pump follows a Weibull distribution with parameters β = 2 and δ = 700 hours. Determine for parts (a) and (b): a. Mean life of a pump b. Variance of the life of a pump c. What is the probability that a pump will last longer tha
> The size of silver particles in a photographic emulsion is known to have a log normal distribution with a mean of 0.001mm and a standard deviation of 0.002 mm. a. Determine the parameter values for the lognormal distribution. b. What is the probability o
> 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 eac
> The time to failure (in hours) for a laser in a cytometry machine is modeled by an exponential distribution with λ = 0.00004. What is the probability that the time until failure is a. At least 20,000 hours? b. At most 30,000 hours? c. Between 20,000 and
> 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. Probability that more than 225 become ill b. Probability
> An article in Electronic Journal of Applied Statistical Analysis [“Survival Analysis of Acute Myocardial Infarction Patients Using Non-Parametric and Parametric Approaches” (2009, Vol. 2(1), pp. 22–36)] described the use of aWeibull distribution to model
> An article in Electric Power Systems Research [“On the Self-Scheduling of a Power Producer in Uncertain Trading Environments” (2008, Vol. 78(3), pp. 311–317)] considered a self-scheduling approach for a power producer. In addition to price and forced out
> An article in Journal of Theoretical Biology [“Computer Model of Growth Cone Behavior and Neuronal Morphogenesis” (1995, Vol. 174(4), pp. 381–389)] developed a model for neuronal morphogenesis in which neuronal growth cones have a significant function in
> Among homeowners in a metropolitan area, 25% recycle paper each week. A waste management company services 10,000 homeowners (assumed independent). Approximate the following probabilities: a. More than 2600 recycle paper in a week b. Between 2400 and 2600
> Provide approximate sketches for beta probability density functions with the following parameters. Comment on any symmetries and show any peaks in the probability density functions in the sketches. a. α = β < 1 b. α = β = 1 c. α = β > 1
> Consider the regional right ventricle transverse wall motion in patients with pulmonary hypertension (PH). The right-ventricle ejection fraction (EF) is approximately normally distributed with a standard deviation of 12 for PH subjects, and with mean and
> The intensity (mW/mm2) of a laser beam on a surface theoretically follows a bivariate normal distribution with maximum intensity at the center, equal variance σ in the x and y directions, and zero covariance. There are several definitions for the width o
> The power in a DC circuit is P = I2/R where I and R denote the current and resistance, respectively. Suppose that I is approximately normally distributed with mean of 200 mA and standard deviation 0.2 mA and R is a constant. Determine the probability den
> Suppose X has a lognormal distribution with parameters θ and ω. Determine the probability density function and the parameters values for Y = Xγ for a constant γ > 0. What is the name of this distribution?
> The continuous uniform random variable X has density function a. Show that the moment-generating function is b. Use MX(t) to find the mean and variance of X.
> Amarketing company performed a risk analysis for a manufacturer of synthetic fibers and concluded that new competitors present no risk 13% of the time (due mostly to the diversity of fibers manufactured), moderate risk 72% of the time (some overlapping o
> The permeability of a membrane used as a moisture barrier in a biological application depends on the thickness of two integrated layers. The layers are normally distributed with means of 0.5 and 1 millimeters, respectively. The standard deviations of lay
> A mechanical assembly used in an automobile engine contains four major components. The weights of the components are independent and normally distributed with the following means and standard deviations (in ounces): a. What is the probability that the we
> Suppose that X and Y have a bivariate normal distribution with σX = 4, σY = 1, μX = 2, μY = 4, and ρ = −0.2. Draw a rough contour plot of the joint probability density function.
> The weight of a small candy is normally distributed with a mean of 0.1 ounce and a standard deviation of 0.01 ounce. Suppose that 16 candies are placed in a package and that the weights are independent. a. What are the mean and variance of the package’s
> The time for an automated system in a warehouse to locate a part is normally distributed with a mean of 45 seconds and a standard deviation of 30 seconds. Suppose that independent requests are made for 10 parts. a. What is the probability that the averag
> Contamination problems in semiconductormanufacturing can result in a functional defect, a minor defect, or no defect in the final product. Suppose that 20%, 50%, and 30% of the contamination problems result in functional, minor, and no defects, respectiv
> The joint distribution of the continuous random variables X, Y, and Z is constant over the region x2 + y2 ≤ 1, 0 < z < 4. Determine the following: a. P(X2 + Y2 ≤ 0.5) b. P(X2 + Y2 ≤ 0.5, Z < 2) c. Joint conditional probability density function of X and
> A continuous random variable X has the following probability distribution: f (x) = 4xe−2x , x > 0 a. Find the moment-generating function for X. b. Find the mean and variance of X.
> Determine the value of c such that the function f (x, y) = cx2y for 0 < x < 3 and 0 < y < 2 satisfies the properties of a joint probability density function. Determine the following: a. P(X < 1, Y < 1) b. P(X < 2.5) c. P(1 < Y < 2.5) d. P(X > 2.1 < Y <
> The percentage of people given an antirheumatoid medication who suffer severe, moderate, or minor side effects are 10, 20, and 70%, respectively. Assume that people react independently and that 20 people are given the medication. Determine the following:
> Show that the following function satisfies the properties of a joint probability mass function: Determine the following: a. P(X c. P(X 0.5, Y e. E(X), E(Y), V(X), V(Y) f. Marginal probability distribution of the random variable X g. Conditional probabil
> The sick-leave time of employees in a firmin 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 ti
> The time it takes a cell to divide (called mitosis) is normally distributed with an average time of 1 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 t
> The probability density function of the time it takes a hematology cell counter to complete a test on a blood sample is f (x) = 0.04 for 50 < x < 75 seconds. a. What percentage of tests requires more than 70 seconds to complete? b. What percentage of tes
> 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(−0.1x) for 0 < x. Determine the probability that a. You arrive by 9:00 A.M. b. You arrive between 8:15 A.M. and 8:30 A.M. c. You arrive
> Determine the cumulative distribution function for the distribution in Exercise 4.1.4.
> Determine the cumulative distribution function for the distribution in Exercise 4.1.2.
> An electron emitter produces electron beams with changing kinetic energy that is uniformly distributed between 3 and 7 joules. Suppose that it is possible to adjust the upper limit of the kinetic energy (currently set to 7 joules). a. What is the mean ki
> The chi-squared random variable with k degrees of freedom has moment-generating function MX(t) = (1 − 2t)−k∕2. Suppose that X1 and X2 are independent chi-squared random variableswith k1 and k2 degrees of freedom, respectively. What is the distribution of
> 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 t
> 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
> 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?
> The thickness of photoresist applied towafers 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
> The thickness of a flange on an aircraft component is uniformly distributed between 0.95 and 1.05 millimeters. Determine the following: a. Cumulative distribution function of flange thickness b. Proportion of flanges that exceeds 1.02 millimeters c. Thic
> A random variable X has the discrete uniform distribution a. Show that the moment-generating function is b. Use MX(t) to find the mean and variance of X.
> Determine the covariance and correlation for the CD4 counts in a month and the following month in Exercise 5.2.6.
> The joint probability distribution is Show that the correlation between X and Y is zero but X and Y are not independent.
> Determine the covariance and correlation for the joint probability density function fXY (x, y) = e−x−y over the range 0 < x and 0 < y.
> Determine the value for c and the covariance and correlation for the joint probability density function fXY (x, y) = cxy over the range 0 < x < 3 and 0 < y < x.
> Assume that 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
> 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)
> Suppose X has a continuous uniformdistribution over the interval [−1, 1]. Determine the following: a. Mean, variance, and standard deviation of X b. Value for x such that P(−x < X < x) = 0.90 c. Cumulative distribution function
> An article in the Journal of Cardiovascular Magnetic Resonance [“Right Ventricular Ejection Fraction Is Better Reflected by Transverse Rather Than Longitudinal Wall Motion in Pulmonary Hypertension” (2010, Vol. 12(35)] discussed a study of the regional r
> A signal in a communication channel is detected when the voltage is higher than 1.5 volts in absolute value. Assume that the voltage is normally distributed with a mean of 0. What is the standard deviation of voltage such that the probability of a false
> An article in Microelectronics Reliability [“Advanced Electronic Prognostics through System Telemetry and Pattern Recognition Methods” (2007, Vol. 47(12), pp. 1865–1873)] presented an example of electronic prognosis. The objective was to detect faults to
> An article in Atmospheric Chemistry and Physics [“Relationship Between Particulate Matter and Childhood Asthma—Basis of a FutureWarning System for Central Phoenix” (2012, Vol. 12, pp. 2479–2490)] reported the use of PM10 (particulate matter
> 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 th
> 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 in
> 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 diam
> The weight of a 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 comp
> 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 devi
> 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 l
> Suppose that the correlation between X and Y is ρ. For constants a, b, c, and d, what is the correlation between the random variables U = aX + b and V = cY + d?
> 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 2 inc
> 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 deviati
> Patients given drug therapy either improve, remain the same, or degrade with probabilities 0.5, 0.4, and 0.1, respectively. Suppose that 20 patients (assumed to be independent) are given the therapy. Let X1, X2, and X3 denote the number of patients who i
> If X and Y have a bivariate normal distribution with ρ = 0, show that X and Y are independent.
> In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Let X and Y denote the milliliters of acid and base needed for equivalence, respectively. Assume that X and Y have a bivariate no
> In the manufacture of electroluminescent lamps, several different layers of ink are deposited onto a plastic substrate. The thickness of these layers is critical if specifications regarding the final color and intensity of light are to be met. Let X and
> Let X and Y represent the concentration and viscosity of a chemical product. Suppose that X and Y have a bivariate normal distribution with σX = 4, σY = 1, μX = 2, and μY = 1. Draw a rough contour plot of the joint probability density function for each o
> Suppose that X and Y have a bivariate normal distribution with σX = 0.04, σY = 0.08, μX = 3.00, μY = 7.70, and ρ = 0. Determine the following: a. P(2.95 < X < 3.05) b. P(7.60 < Y < 7.80) c. P(2.95 < X < 3.05, 7.60 < Y < 7.80)
> A Web site uses ads to route visitors to one of four landing pages. The probabilities for each landing page are equal. Consider 20 independent visitors and let the random variablesW, X, Y, and Z denote the number of visitors routed to each page. Calculat
> Based on the number of voids, a ferrite slab is classified as either high, medium, or low. Historically, 5% of the slabs are classified as high, 85% as medium, and 10% as low. Agroup of 20 slabs that are independent regarding voids is selected for testin
> Determine the covariance and correlation for the lengths of the minor and major axes in Exercise 5.2.5.
> 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 mg/dl. The Food and Nutrition Institute of the Philippines found that the total cholestero
> 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
> 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 ar
> An article in Knee Surgery Sports Traumatology Arthroscopy [“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 cr
> 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 < X < 10) = 0 d. P(−x < X − 10 < x) = 0.95 e. P(−x < X − 10
> Assume that X is normally distributed with a mean of 10 and a standard deviation of 2. Determine the following: a. P(Z < 13) b. P(Z > 9) c. P(6 < X < 14) d. P(2 < X < 4) e. P(−2 < X < 8)
> A set of 200 independent patients take antiacid medication at the start of symptoms, and 80 experience moderate to substantial relief within 90 minutes. Historically, 30% of patients experience relief within 90 minutes with no medication. If the medicati
> An article in Financial Markets Institutions and Instruments [“Pricing Reinsurance Contracts on FDIC Losses” (2008, Vol. 17(3), pp. 225–247)] modeled average annual losses (in billions of dollars) of the Federal Deposit Insurance Corporation (FDIC) with
> An article in IEEE Transactions on Dielectrics and Electrical Insulation [“Statistical Analysis of the AC Breakdown Voltages of Ester Based Transformer Oils” (2008, Vol. 15(4), pp. 1044–1050)] used Weibull distributions to model the breakdown voltage of
> An article in Electronic Journal of Applied Statistical Analysis [“Survival Analysis of Dialysis Patients Under Parametric and Non-Parametric Approaches” (2012, Vol. 5(2), pp. 271–288)] modeled the survival time of dialysis patients with chronic kidney d
> An article in Sensors and Actuators A: Physical [“Characterization and Simulation of Avalanche PhotoDiodes for Next-Generation Colliders” (2011, Vol. 172(1), pp. 181–188)] considered an avalanche photodiode (APD) to detect charged particles in a photo. T
> The geometric random variable X has probability distribution f (x) = (1 − p)x−1p, x = 1, 2,… a. Show that the moment-generating function is b. Use MX(t) to find the mean and variance of X.
> An article in Proceedings of the 33rd International ACM SIGIR Conference on Research and Development in Information Retrieval [“Understanding Web Browsing Behaviors Through Weibull Analysis of Dwell Time” (2010, pp. 379–386)] proposed that a Weibull dist
> Suppose that X has a Weibull distribution with β = 2 and δ = 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
> Suppose that the lifetime of a component (in hours) is modeled with a Weibull distribution with β = 2 and δ = 4000. Determine the following in parts (a) and (b): a. P(X > 5000) b. P(X > 8000 | X > 3000) c. Comment on the probabilities in the previous par
> An article in the Journal of the Indian Geophysical Union titled “Weibull and Gamma Distributions for Wave Parameter Predictions” (2005, Vol. 9, pp. 55–64) described the use of the Weibull distribution to model ocean wave heights. Assume that the mean wa
> An article in the Journal of Geophysical Research [“Spatial and Temporal Distributions of U.S. Winds and Wind Power at 80 m Derived from Measurements” (2003, Vol. 108)] considered wind speed at stations throughout the United States. For a station at Amar
