Questions and Answers of Statistics For Engineering And The Sciences

The Ohio Environmental Protection Agency used the Index of Biotic Integrity (IBI) to measure the biological condition or health of an aquatic region. The IBI is the sum of metrics which measure the
Let Y1 and Y2 be a sample of n = 2 observations from a gamma random variable with parameters α = 1 and arbitrary β, and corresponding density functionShow that the sum W = (Y1 + Y2) is also a gamma
Let X and Y be two continuous random variables with joint probability densitya. Find the value of c.b. Find f1(x).c. Find f2(y).d. Find f1 (x Ι y).e. Find f2 (x Ι y).f. Find P (X ≤ 1 and Y ≤
Let X and Y be two continuous random variables with joint probability density f(x, y). The joint distribution function F(a, b) is defined as follows:Verify each of the following:a. F (-∞, -∞) =
Refer to the IEEE Transactions on Information Forensics and Security (March 2013) study of wireless identify theft using cloned credit or debit cards, Exercise 6.3. On average, how many genuine balls
Refer to the Canadian Journal of Civil Engineering (Jan. 2013) study, Exercise 6.5.Data from Exercise 6.5Refer to the Canadian Journal of Civil Engineering (Jan. 2013) investigation of the use of
Refer to Exercise 6.7.Data from Exercise 6.7A special delivery truck travels from point A to point B and back over the same route each day. There are three traffic lights on this route. Let X be the
Refer to Exercise 6.12.a. Find E (Y - 10).b. Find E(3Y).Data from Exercise 6.12The Department of Transportation (DOT) monitors sealed bids for new road construction. For new access roads in a certain
Refer to Exercise 6.16.a. Find E(X).b. Find E(Y).c. Find E (X + Y).d. Find E(XY).Data from Exercise 6.16Let X and Y have the joint density Scxy if 0 s x < 1; 0 s ys 1 lo f(x, y) elsewhere
Refer to Exercise 6.17.a. Find E(X).b. Find E(Y).c. Find E (X + Y).d. Find E(XY).Data from Exercise 6.17Let X and Y have the joint density fx + cy if 1 sIS 2;0 s y
Let X and Y be two continuous random variables with joint probability distribution f(x, y). Consider the function g(X). Show that 00 E[g(X)] = 8(x)fi(x) dx
Prove Theorems 6.1–6.3 for discrete random variables X and Y. THEOREM 6.1 Let c be a constant. Then the expected value of c is E(c) = c
Prove Theorems 6.1–6.3 for continuous random variables X and Y. THEOREM 6.1 Let c be a constant. Then the expected value of c is E(c) = c
Find the covariance of the random variables X and Y in Exercise 6.2.Data from Exercise 6.2Consider the experiment of tossing a pair of dice. Let X be the outcome (i.e., the number of dots appearing
As an illustration of why the converse of Theorem 6.6 is not true, consider the joint distribution of two discrete random variables, X and Y, shown in the accompanying table. Show that Cov (X, Y) =
Use Theorem 6.7 to draw a random sample of n = 5observations from a distribution with probability density function Se if y < 0 elsewhere
Use Theorem 6.7 to draw a random sample of n = 5 observations from a beta distribution with α = 2 and β = 1. THEOREM 6.7 Let Y be a continuous random variable with density function f( y) and
The total time X (in minutes) from the time a supercomputer job is submitted until its run is completed and the time Y the job waits in the job queue before being run have the joint density
A large freight elevator can transport a maximum of 10,000 pounds (5 tons). Suppose a load of cargo containing 45 boxes must be transported via the elevator. Experience has shown that the weight Y of
Refer to the Earthquake Engineering and Engineering Vibration (March 2013) study of the structural damage to a three-story building caused by seismic ground noise, Exercise 5.33. Recall that the
Refer to the Electrical Engineering (March 2013) assessment of the quality of electrical power, Exercise 5.37. Recall that a measure of quality is the degree to which voltage fluctuations cause light
Let Y1 Y2, . . . , Yn1, be a random sample of n1 observations from a normal distribution with mean μ1 and variance σ12. Let X1, X2, . . . , Xn2 be a random sample of n2 observations from a normal
Let S12 and S22 be the variances of independent random samples of sizes n1 and n2 selected from normally distributed populations with parameters (μ1, σ2) and (μ2, σ2), respectively. Thus, the
Refer to the Journal of Engineering for Industry (Aug. 1993) study of an automated drilling machine, Exercise 3.67. The eight machining conditions used in the study are reproduced here.Suppose that
Refer to the U.S. Army Corps of Engineers data on contaminated fish saved in the DDT file. Recall that the length (in centimeters), weight (in grams), and DDT level (in parts per million) were
The management of a bank must decide whether to install a commercial loan decision-support system (an online management information system) to aid its analysts in making commercial loan decisions.
Suppose that X and Y, the proportions of an 8-hour workday that two quality control inspectors actually spend on performing their assigned duties, have joint probability densitya. Find the marginal
Studies by neuroscientists at the Massachusetts Institute of Technology (MIT) reveal that melatonin, which is secreted by the pineal gland in the brain, functions naturally as a sleep-inducing
The merging process from an acceleration lane to the through lane of a freeway constitutes an important aspect of traffic operation at interchanges. A study of parallel interchange ramps in Israel
Concrete experiences a characteristic marked increase in “creep” when it is heated for the first time under load. An experiment was conducted to investigate the transient thermal strain behavior
A supplier of home heating oil has a 250-gallon tank that is filled at the beginning of each week. Since the weekly demand for the oil increases steadily up to 100 gallons and then levels off between
Dioxin, often described as the most toxic chemical known, is created as a by-product in the manufacture of herbicides such as Agent Orange. Scientists have found that .000005 gram (five-millionths of
The determination of the percent canopy closure of a forest is essential for wildlife habitat assessment, watershed runoff estimation, erosion control, and other forest management activities. One way
Refer to Exercise 6.113. Let S2 be the variance of the TMS readings for the 32 sampled forest sites. Assuming the sample is from a normal population, estimate the probability that S2 exceeds
Quality control is a problem with items that are mass-produced. The production process must be monitored to ensure that the rate of defective items is kept at an acceptably low level. One method of
IEEE Transactions (June 1990) presented a hybrid algorithm for solving polynomial 0–1 mathematical programming problems. The solution time (in seconds) for a randomly selected problem solved using
An article in Industrial Engineering (August 1990) discussed the importance of modeling machine downtime correctly in simulation studies. As an illustration, the researcher considered a
Let X and Y be uncorrelated random variables. Verify each of the following:a. V(X + Y) = V(X - Y)b. Cov[(X + Y), (X - Y)] = V(X) - V(Y)
Suppose three continuous random variables Y1, Y2, Y3 have the joint distributiona. Find the value of c that makes f(y1, y2, y3) a probability density.b. Are the three variables independent?
Consider the density functionFind the density function of W, where:a. W = √Yb. W = 3 - Yc. W = - ln (Y) S3y if 0 < ys 1 if 0 s y s 1 f(y) to elsewhere
Let Y have an exponential density with mean β. Show that W = 2Y/β has a x2 density with v = 2 degrees of freedom.
Suppose that Y1, Y2, and Y3 are random variables with (μ1 = 0, σ12 = 2), (μ2 = 1, σ22 = 3), (μ3 = 5, σ32 = 9), Cov (Y1, Y2) = 1, Cov (Y1, Y3) = 4, and Cov (Y2, Y3) = -2Find the mean
Suppose that Y1, Y2, Y3, and Y4 are random variables withFind the mean and variance ofl = -3Y1 + 2Y2 + 6Y3 - Y4 E(Y¡) = 2, V(Y1) = 4, Cov(Y1, Y2) = -1 Cov(Y2, Y3) = 0 E(Y,) = 4, V(Y2) = 8, Cov(Y,,
Let Y1, Y2, . . . , Yn be a sample of n independent observations selected from a gamma distribution with α = 1 and β = 2 . Show that the expected value and variance of the sample mean Y̅ are
Consider the density functionFind the density function of W, where:a. W = e-Yb. W = Y - 3c. W = Y/3 fe-o-3) to if y > 3 f(y) %3D elsewhere
Consider the density functionFind the density function of W, where:a. W = Y2b. W = 2Y -1c. W = 1/Y S2y if 0 s y s 1 lo elsewhere (v)
Refer to the American Mineralogist (October 2009) study of the evolution of uranium minerals in the Earth’s crust, Exercise 5.17. Recall that researchers estimate that the trace amount of uranium Y
The National Institute for Occupational Safety and Health (NIOSH) evaluated the level of exposure of workers to the chemical dioxin, 2,3,7,8-TCDD. The distribution of TCDD levels in parts per
Use Monte Carlo simulation to approximate the sampling distribution of s2, the variance of a sample of n = 100 observations from a.a. Uniform distribution on the interval (0, 1).b. Normal
Use Monte Carlo simulation to approximate the sampling distribution of M, the median of a sample of n = 50 observations from a uniform distribution on the interval (0, 1).
Use Monte Carlo simulation to approximate the sampling distribution of R, the range of a sample of n = 10 observations from a normal distribution, with mean 0 and variance 1.
Refer to the International Conference on Social Robotics (Vol. 6414, 2010), study of the trend in the design of social robots, Exercise 2.1. The researchers obtained a random sample of 106 social
The British Medical Journal (August 17, 2002) published a study to compare the effectiveness of handwashing with soap and hand rubbing with alcohol. Health care workers who used hand rubbing had a
The U.S. Army Engineering and Housing Support Center recently sponsored a study of the reliability, availability, and maintainability (RAM) characteristics of small diesel and gas-powered systems at
The computer chips in notebook and laptop computers are produced from semiconductor wafers. Certain semiconductor wafers are exposed to an environment that generates up to 100 possible defects per
According to the International Association of Women in Fire and Protection Services, 4% of all fire fighters in the world are female.a. Approximate the probability that more than 100 of a random
If Y has a x2 distribution with n degrees of freedom then Y could be represented by ∑x = in xi, where the xi's are independent x2 distributions, each with degree of freedom.a. Show that Z =
Let p̂1 be the sample proportion of successes in a binomial experiment with n1 trials and let p̂2 be the sample proportion of successes in a binomial experiment with n2 trials, conducted
Refer to the Journal of Agricultural, Biological and Environmental Sciences (Sep. 2000) evaluation of the risk posed by hazardous pollutants, Exercise 4.28. In the experiment, guppies (all the same
Consumer Reports found widespread contamination of seafood in New York and Chicago supermarkets. For example, 40% of the swordfish pieces available for sale have a level of mercury above the Food and
Refer to the Scientific American (July 2003) report on whether bottled water is really purified water, Exercise 4.29. Recall that the Natural Resources Defense Council found that 25% of bottled water
Refer to the Journal of Performance of Constructed Facilities (Feb. 2005) study of inspection ratings of all major Denver bridges, Exercise 4.30. Recall that the National Bridge Inspection Standard
New Jersey Business reports that Newark International Airport’s terminal handles an average of 3,000 international passengers an hour but is capable of handling twice that number. Also, after
Refer to the Transactions of the ASME (June 2004) study on predicting daily natural gas consumption using temperature, Exercise 5.32. Recall that the researchers showed that the daily July
Refer to Exercise 6.95. Find the exact probabilities using statistical software.Data from Exercise 6.95Refer to the Transactions of the ASME (June 2004) study on predicting daily natural gas
Refer to the IEICE Transactions on Information & Systems (Jan. 2005) study of impedance to leg movements, Exercise 2.46. Recall that engineers attached electrodes to the ankles and knees of
Refer to the Composites Fabrication Magazine (Sept. 2004) evaluation of a new method of fastening fiber-reinforced polymer (FRP) strips to concrete, Exercise 2.47. Recall that a sample of 10 FRP
Refer to Exercise 6.1. Are X and Y independent?Data from Exercise 6.1A software program is designed to perform two tasks, A and B. Let X represent the number of IF-THEN statements in the code for
Refer to Exercise 6.2. Are X and Y independent?Data from Exercise 6.2Consider the experiment of tossing a pair of dice. Let X be the outcome (i.e., the number of dots appearing face up) on the first
Refer to Exercise 6.3. Are X and Y independent?Data from Exercise 6.3Refer to the IEEE Transactions on Information Forensics and Security (March 2013) study of wireless identify theft using cloned
Refer to Exercise 6.6. Are X and Y independent?Data from Exercise 6.6Refer to The International Journal of Robotics Research (Dec. 2004) study of a robot-sensor system in an unknown environment. In
Refer to the Journal of Systems Sciences & Systems Engineering (March 2013) study of the reliability of a manufacturing system for producing integrated circuit (IC) cards that involves two
In the Journal of Hydrological Sciences (April 2000), the bivariate normal distribution was used to model the joint distribution of annual storm peak (i.e., maximum rainfall intensity) and total
The life length Y (in hundreds of hours) for fuses used in a tele video computer terminal has an exponential distribution with mean β = 5. Each terminal requires two such fuses—one acting as a
Let X and Y denote the lifetimes of two different types of components in an electronic system. The joint density of X and Y is given byShow that X and Y are independent. [Hint: A theorem in
Refer to Exercise 6.15. Show that X and Y are independent.Data from Exercise 6.15Refer to the Journal of Engineering for Industry (May 1993) study of friction feed paper separation. Consider a system
Refer to Exercise 6.16. Are X and Y independent?Data from Exercise 6.16Let X and Y have the joint density Scxy if 0 < x < 1; 0 s y s 1 lo f(x, y) elsewhere
Refer to Exercise 6.17. Are X and Y independent?Data from Exercise 6.17Let X and Y have the joint density Sx + cy if 1 s xs 2; 0 s ys 1 f(x, y) = elsewhere
Prove Theorem 6.4 for the continuous case. THEOREM 6.4 If X and Y are independent random variables, then E(XY) = E(X)E(Y) Proof of Theorem 6.4 We will prove the theorem for the discrete case. The
Find the covariance of the random variables X and Y in Exercise 6.1.Data from Exercise 6.1A software program is designed to perform two tasks, A and B. Let X represent the number of IF-THEN
Find the correlation coefficient r for X and Y in Exercise 6.5.Data from Exercise 6.5Refer to the Canadian Journal of Civil Engineering (Jan. 2013) investigation of the use of variable speed limits
Refer to Exercise 6.7.a. Find the covariance of the random variables x and y.b. Find the coefficient of correlation r for x and y.Data from Exercise 6.7A special delivery truck travels from point A
Commercial kerosene is stocked in a bulk tank at the beginning of each week. Because of limited supplies, the proportion X of the capacity of the tank available for sale and the proportion Y of the
Refer to the Journal of Hydrological Sciences (April 2000) study of rainfall in Tokushima, Japan, Exercise 6.34. Recall that the bivariate normal distribution was used to model the joint distribution
Refer to Exercise 6.16.a. Find the covariance of the random variables X and Y.b. Find the coefficient of correlation ρ for X and Y.Data from Exercise 6.16Let X and Y have the joint density f(x, y) =
Refer to Exercise 6.17.a. Find the covariance of the random variables X and Y.b. Find the coefficient of correlation ρ for X and YData from Exercise 6.17Let X and Y have the joint density Sx + cy if
Prove Theorem 6.6 for the discrete case. THEOREM 6.6 If two random variables X and Y are independent, then Cov(X, Y) = 0
Prove Theorem 6.6 for the continuous case. THEOREM 6.6 If two random variables X and Y are independent, then Cov(X, Y) = 0
Find the covariance of X and Y for the random variables of Exercise 6.18.Data from Exercise 6.18Let X and Y be two continuous random variables with joint probability density (ce-(r+y) if 0sx< 00; 0 s
Refer to Exercise 6.2. Find the mean and variance of (X + Y) , the sum of the dots showing on the two dice.Data from Exercise 6.2Consider the experiment of tossing a pair of dice. Let X be the
Refer to Exercise 6.7. Find the variance of (X + Y) . Within what range would you expect (X + Y) to fall?Data from Exercise 6.7A special delivery truck travels from point A to point B and back over
Refer to Exercise 6.14. Find the variance of (X - Y), the time it takes to actually service the car.Data from Exercise 6.14The joint density of X, the total time (in minutes) between an
Refer to the Journal of Engineering Mechanics (Dec. 2009) study of the characteristics of a 10-bar truss subjected to loads, Exercise 6.13. Recall that the joint probability distribution of stiffness
A particular manufacturing process yields a proportion p of defective items in each lot. The number Y of defectives in a random sample of n items from the process follows a binomial distribution.
Researchers at the University of Kent (England) developed models for counting the number of colonies of microorganisms in liquid (Journal of Agricultural, Biological, and Environmental Statistics,
The amount Y of paper used per day by a laser printer at a commercial copy center has an exponential distribution with mean equal to five boxes (i.e., β = 5). The daily cost of the paper is
An environmental engineer has determined that the amount Y (in parts per million) of pollutant per water sample collected near the discharge tubes of an island power plant has probability density
Researchers at the University of California (Berkeley) have developed a switched-capacitor circuit for generating pseudorandom signals (International Journal of Circuit Theory and Applications,
Let X and Y have the joint densitywhere c is a constant.a. Find the value of c that makes f(x, y) a probability density function.b. Find the marginal density for y and showc. Find f1 (x Ι y), the
Let X and Y have the joint densitya. Find the value of c that makes f(x, y) a probability density function.b. Find the marginal densities f1(x) and f2(y).c. Find the conditional densities f1 (x Ι y)

Showing 700 - 800 of 1103