Determine the mean and variance of the random variable in Exercise 4.1.10.
> Hits to a high-volumeWeb site are assumed to follow a Poisson distribution with a mean of 10,000 per day. Approximate each of the following: a. Probability of more than 20,000 hits in a day. b. Probability of less than 9900 hits in a day. c. Value such t
> The manufacturing of semiconductor chips produces 2% defective chips. Assume that the chips are independent and that a lot contains 1000 chips. Approximate the following probabilities: a. More than 25 chips are defective. b. Between 20 and 30 chips are d
> 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 pers
> Suppose that X is a Poisson random variable with λ = 6. a. Compute the exact probability that X is less than four. b. Approximate the probability that X is less than four and compare to the result in part (a). c. Approximate the probability that 8 < X <
> The number of (large) inclusions in cast iron follows a Poisson distribution with a mean of 2.5 per cubic millimeter. Approximate the following probabilities: a. Determine the mean and standard deviation of the number of inclusions in a cubic centimeter
> Cabs pass your workplace according to a Poisson process with a mean of five cabs per hour. a. Determine the mean and standard deviation of the number of cabs per 10-hour day. b. Approximate the probability that more than 65 cabs pass within a 10-hour day
> Suppose that X is a binomial random variable with n = 200 and p = 0.4. Approximate the following probabilities: a. P(X ≤ 70) b. P(70 < X < 90) c. P(X = 80)
> In Exercise 5.2.7, the monthly demand for MMR vaccine was assumed to be approximately normally distributed with a mean and standard deviation of 1.1 and 0.3 million doses, respectively. Suppose that the demands for different months are independent, and l
> Patients arrive at a hospital emergency department according to a Poisson process with a mean of 6.5 per hour. a. What is the mean time until the 10th arrival? b. What is the probability that more than 20 minutes is required for the third arrival?
> The rate of return of an asset is the change in price divided by the initial price (denoted as r). Suppose that $10,000 is used to purchase shares in three stocks with rates of returns X1, X2, X3. Initially, $2500, $3000, and $4500 are allocated to each
> A U-shaped component is to be formed from the three parts A, B, and C. See Figure 5.14. The length of A is normally distributed with a mean of 10 mm and a standard deviation of 0.1 mm. The thicknesses of parts B and C are each normally distributed with a
> Assume that the weights of individuals are independent and normally distributed with a mean of 160 pounds and a standard deviation of 30 pounds. Suppose that 25 people squeeze into an elevator that is designed to hold 4300 pounds. a. What is the probabil
> An automated filling machine fills soft-drink cans, and the standard deviation is 0.5 fluid ounce. Assume that the fill volumes of the cans are independent, normal random variables. a. What is the standard deviation of the average fill volume of 100 cans
> 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 ACL recons
> Making handcrafted pottery generally takes two major steps: wheel throwing and firing. The time of wheel throwing and the time of firing are normally distributed random variables with means of 40 minutes and 60 minutes and standard deviations of 2 minute
> 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
> Suppose that the random variable X represents the length of a punched part in centimeters. Let Y be the length of the part inmillimeters. If E(X) = 5 and V(X) = 0.25, what are the mean and variance of Y?
> X and Y are independent, normal random variables with E(X) = 2, V(X) = 5, E(Y) = 6, and V(Y) = 8. Determine the following: a. E(3X + 2Y) b. V(3X + 2Y) c. P(3X + 2Y < 18) d. P(3X + 2Y < 28)
> 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 that the messages arrive at the node according to a Poisson process with λ = 30 messages per minute. Fiv
> For the Transaction Processing Performance Council’s benchmark in Exercise 5.1.6, let X, Y, and Z denote the average number of selects, updates, and inserts operations required for each type of transaction, respectively. Calculate the following: a. Corre
> Patients are given a drug treatment and then evaluated. Symptoms either improve, degrade, or remain the same with probabilities 0.4, 0.1, 0.5, respectively. Assume that four independent patients are treated and let X and Y denote the number of patients w
> Determine the value for c and the covariance and correlation for the joint probability mass function fXY (x, y) = c(x + y) for x = 1, 2, 3 and y = 1, 2, 3.
> Determine the covariance and correlation for the following joint probability distribution:
> Determine the cumulative distribution function for the distribution in Exercise 4.1.3.
> Determine the cumulative distribution function for the distribution in Exercise 4.1.1.
> Suppose that the cumulative distribution function of the random variable X is Determine the following: a. P(X b. P(X > 1.5) c. P(X d. P(X > 6)
> Determine the cumulative distribution function for the random variable in Exercise 4.1.8. Use the cumulative distribution function to determine the probability that 400 < X < 500.
> Determine the cumulative distribution function for the random variable in Exercise 4.1.9. Use the cumulative distribution function to determine the probability that the waiting time is less than 1 hour.
> Determine the cumulative distribution function for the random variable in Exercise 4.1.10. Use the cumulative distribution function to determine the probability that the random variable is less than 55.
> 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 unt
> Consider the joint distribution in Exercise 5.1.10. Determine the following: a. fY | 2(y) b. E(Y | X = 2) c. V(Y | X = 2) d. Are X and Y independent?
> Determine the value of c that makes the function f (x, y) = c(x + y) a joint probability density function over the range 0 < x < 3 and x < y < x + 2. Determine the following: a. P(X < 1, Y < 2) b. P(1 < X < 2) c. P(Y > 1) d. P(X < 2, Y < 2) e. E(X) f.
> An article in Chemosphere [“Statistical Evaluations Reflecting the Skewness in the Distribution of TCDD Levels in Human Adipose Tissue” (1987, Vol. 16(8), pp. 2135–2140)] concluded that the levels of 2,3,7,8-TCDD (colorless persistent environmental conta
> An article in AppliedMathematics and Computation [“Confidence Intervals for Steady State Availability of a System with Exponential Operating Time and Lognormal Repair Time” (2003, Vol. 137(2), pp. 499–509)] considered the long-run availability of a syste
> An article in Journal of Hydrology [“Use of a Lognormal Distribution Model for Estimating Soil Water Retention Curves from Particle-Size Distribution Data” (2006, Vol. 323(1), pp. 325–334)] considered a lognormal distribution model to estimate water rete
> Suppose that the length of stay (in hours) at a hospital emergency department is modeled with a lognormal random variable X with θ = 1.5 and ω = 0.4. Determine the following in parts (a) and (b): a. Mean and variance b. P(X < 8) c. Comment on the differe
> An article in Health and Population: Perspectives and Issues (2000, Vol. 23, pp. 28–36) used the lognormal distribution to model blood pressure in humans. The mean systolic blood pressure (SBP) in males age 17 was 120.87 mm Hg. If the coefficient of vari
> 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 probabil
> Suppose that X has a lognormal distribution and that the mean and variance of X are 100 and 85,000, respectively. Determine the parameters θ and ω2 of the lognormal distribution. [Hint: Define u = exp(θ) and v = exp(ω2) and write two equations in terms o
> 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 θ = 0.5 and ω2 = 1. a. What is the probability that a page is viewed for more than 10 seconds? b. By what
> Rawmaterials 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
> Suppose that X has a lognormal distribution with parameters θ = 2 and ω2 = 4. Determine the following in parts (a) and (b): a. P(X < 500) b. Conditional probability that X < 1500 given that X > 1000 c. What does the difference between the probabilities i
> Suppose that X has a lognormal distribution with parameters θ = 5 and ω2 = 9. Determine the following: a. P(X < 13,300) b. Value for x such that P(X ≤ x) = 0.95 c. Mean and variance of X
> Determine the mean and variance of the random variable in Exercise 4.1.8.
> Determine the mean and variance of the random variable in Exercise 4.1.1.
> Suppose that f (x) = 0.125x for 0 < x < 4. Determine the mean and variance of X.
> Suppose that f (x) = 1.5x2 for −1 < x < 1. Determine the mean and variance of X.
> The thickness of a conductive coating in micrometers has a density function of 600x−2 for 100 μm < x < 120 μm. 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 th
> 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 < x < 1205 millimeters
> Suppose that the random variables X, Y, and Z have the joint probability density function fXYZ(x, y, z) = c over the cylinder x2 + y2 < 4 and 0 < z < 4. Determine the constant c so that fXYZ(x, y, z) is a probability density function. Determine the follo
> Calls to a telephone system follow a Poisson process 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 10th call? b. What is the mean time until the 10th call? c. What is the
> Suppose that the random variables X, Y, and Z have the joint probability density function f (x, y, z) = 8xyz for 0 < x < 1, 0 < y < 1, and 0 < z < 1. Determine the following: a. P(X < 0.5) b. P(X < 0.5, Y < 0.5) c. P(Z < 2) d. P(X < 0.5 or Z < 2) e. E(
> Suppose that the random variables X, Y, and Z have the following joint probability distribution. Determine the following: a. P(X = 2) b. P(X = 1, Y = 2) c. P(Z d. P(X = 1 or Z = 2) e. E(X) f. P(X = 1 | Y = 1) g. P(X = 1, Y = 1 | Z = 2) h. P(X = 1 | Y
> For the Transaction Processing Performance Council’s benchmark in Exercise 5.1.6, let X, Y, and Z denote the average number of selects, updates, and inserts operations required for each type of transaction, respectively. Calculate the following: a. fXYZ(
> The systolic and diastolic blood pressure values (mm Hg) are the pressures when the heart muscle contracts and relaxes (denoted as Y and X, respectively). Over a collection of individuals, the distribution of diastolic pressure is normal with mean 73 and
> An article in Clinical Infectious Diseases [“Strengthening the Supply of Routinely Administered Vaccines in the United States: Problems and Proposed Solutions” (2006, Vol. 42(3), pp. S97–S103)] reported that recommended vaccines for infants and children
> An article in Health Economics [“Estimation of the TransitionMatrix of a Discrete-TimeMarkov Chain” (2002, Vol. 11, pp. 33–42)] considered the changes in CD4 white blood cell counts from one month to
> The lengths of the minor and major axes are used to summarize dust particles that are approximately elliptical in shape. Let X and Y denote the lengths of theminor andmajor axes (inmicrometers), respectively. Suppose that fX(x) = exp (−x), 0 < x and the
> A manufacturer of electroluminescent lamps knows that the amount of luminescent ink deposited on one of its products is normally distributed with a mean of 1.2 grams and a standard deviation of 0.03 gram. Any lamp with less than 1.14 grams of luminescent
> Consider the joint distribution in Exercise 5.1.4. Determine the following: a. fY | 3(y) b. E(Y | X = 3) c. V(Y | X = 3)
> Consider the joint distribution in Exercise 5.1.1. Determine the following: a. Conditional probability distribution of Y given that X = 1.5 b. Conditional probability distribution of X that Y = 2 c. E(Y | X = 1.5) d. Are X and Y independent?
> Given the probability density function f (x) = 0.013x2e−0.01x/Γ(3), determine the mean and variance of the distribution.
> The conditional probability distribution of Y given X = x is fY|x(y) = xe−xy for y > 0, and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10. a. Graph fY|x(y) = xe−xy for y > 0 for several values of x. Determin
> Thewaiting time for service at a hospital emergency department (in hours) follows a distribution with probability density function f (x) = 0.5 exp(−0.5x) for 0 < x.Determine the following: a. P(X < 0.5) b. P(X > 2) c. Value x (in hours) exceeded with pr
> The probability density function of the length of a metal rod is f (x) = 2 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 proba
> 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
> 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 1 hour for a taxi? b. Suppose that you have already been waiting for 1 hour for a taxi.
> 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
> 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 c
> The life of automobile voltage regulators has an exponential distribution with a mean life of 6 years. You purchase a 6-year-old automobile with a working voltage regulator and plan to own it for 6 years. a. What is the probability that the voltage regul
> Suppose that the log-ons to a computer network follow a Poisson process with an average of three 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 probab
> Suppose that X has an exponential distribution with a mean of 10. Determine the following: a. P(X < 5) b. P(X < 15 | X > 10) c. Compare the results in parts (a) and (b) and comment on the role of the lack of memory property.
> 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 λ and r of the distribution. b. Assume that each step has the same distribution for service
> Suppose that X has an exponential distribution with mean equal to 10. Determine the following: a. P(X > 10) b. P(X > 20) c. P(X < 30) d. Find the value of x such that P(X < x) = 0.95.
> An article in Vaccine [“Modeling the Effects of Influenza Vaccination of Health Care Workers in Hospital Departments” (2009, Vol. 27(44), pp. 6261–6267)] considered the immunization of healthcare workers to reduce the hazard rate of influenza virus infec
> An article in Ad Hoc Networks [“Underwater Acoustic Sensor Networks: Target Size Detection and Performance Analysis” (2009, Vol. 7(4), pp. 803–808)] discussed an underwater acoustic sensor network to monitor a given area in an ocean. The network does not
> An article in the Journal of the National Cancer Institute [“Breast Cancer Screening Policies in Developing Countries: A Cost-Effectiveness Analysis for India” (2008, Vol. 100(18), pp. 1290–1300)] presented a screening analysis model of breast cancer bas
> The length of stay at a specific emergency department in a hospital 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 probabi
> 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 completion time for the first phase is modeled with a beta distribution and the minimum,
> An allele is an alternate form of a gene, and the proportion of alleles in a population is of interest in genetics. An article in BMC Genetics [“Calculating Expected DNA Remnants from Ancient Founding Events in Human Population Genetics” (2008, Vol. 9, p
> 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 α = 2 and β = 3. What is the probability that the task requires more than two days to complete?
> The length of stay at a hospital 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 α = 10 and β = 1. Determine the following: a. P(X > 0.9) b. P(X < 0.5
> 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,
> Use the properties of the gamma function to evaluate the following: a. Γ(6) b. Γ(5/2) c. Γ(9/2)
> Suppose that X has a beta distribution with parameters α = 2.5 and β = 2.5. Sketch an approximate graph of the probability density function. Is the density symmetric?
> Suppose that x has a beta distribution with parameters α = 2.5 and β = 1. Determine the following: a. P(X < 0.25) b. P(0.25 < X < 0.75) c. Mean and variance
> If the random variable X has an exponential distribution with mean θ, determine the following: a. P(X > θ) b. P(X > 2θ) c. P(X > 3θ) d. How do the results depend on θ?
> Derive the formula for the mean and variance of an exponential random variable.
> 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 one-half hour
> 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
> 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 that the number of fragments follows a Poisson distribution. a. What is the mean number of grams of chocolate until a
> Powermeters enable cyclists to obtain power measurements nearly continuously. The meters also calculate the average power generated over a time interval. Professional riders can generate 6.6 watts per kilogram of body weight for extended periods of time.
> Derive the probability density function for a lognormal random variable Y from the relationship that Y = exp(W) for a normal random variable W with mean θ and variance ω2.
> An article in Mathematical Biosciences [“Influence of Delayed Viral Production on Viral Dynamics in HIV-1 Infected Patients” (1998, Vol. 152(2), pp. 143–163)] considered the time delay between the initial infection by immunodeficiency virus type 1 (HIV-1
> A random variable X has the Poisson distribution a. Show that the moment-generating function is b. Use MX(t) to find the mean and variance of the Poisson random variable.
> Jennifer Olde calls you requesting an explanation of the fact-finding determination of a Federal Court of Appeals. Prepare a letter to be sent to Jenni fer answering this query. Her address is 3246 Highland Dri ve, Clifton, VA 20124.
> Interpret each of the following citations. a. Temp.Reg.§ 1.707-5T(a)(2). b. Rev.Rul. (l()-11, 1960-1 C.B. 174 c. TAM 8837003.
> Where can a researcher find newly issued Proposed, final, and Temporary Regulations?
> Why are certain Code Section numbers missing from the Internal Revenue Code (e.g., §§ 6, 7, 8, 9. 10)?
> Is § 212(1) a proper Code Section citation? Why or why not?
