Questions and Answers of Statistics For Engineering And The Sciences

Refer to the Journal of Engineering for Industry (May 1993) study of friction feed paper separation, Exercise 5.12. Consider a system that utilizes two interrelated feed paper separators. The joint
The joint density of X, the total time (in minutes) between an automobile’s arrival in the service queue and its leaving the system after servicing, and Y, the time (in minutes) the car waits in
In building construction, a truss is a structure comprised of triangular units whose ends are connected at joints referred to as nodes. The Journal of Engineering Mechanics (Dec. 2009) published a
The Department of Transportation (DOT) monitors sealed bids for new road construction. For new access roads in a certain state, let X = low bid (thousands of dollars) and let Y = DOT estimate
Let X and Y be two discrete random variables with joint probability distribution p(x, y). DefineF1(a) = P(X ≤ a) and F1 (a Ι y) = P(X ≤ a Ι y)Verify each of the following:a.b. F1(a) =
The joint probability distribution for two discrete random variables, X and Y, is given by the formulaVerify that the properties of a bivariate probability distribution are satisfied. p(x, y) =
The Face Recognition Technology (FERET) program, sponsored by the U.S. Department of Defense, was designed to develop automatic face recognition capabilities to assist homeland security. A biometric
From a group of three data processing managers, two senior systems analysts, and two quality control engineers, three people are to be randomly selected to form a committee that will study the
A 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 number of red lights the truck encounters on
Refer to The International Journal of Robotics Research (Dec. 2004) study of a robot-sensor system in an unknown environment, Exercise 4.7. In the three-point, single-link robotic system shown in the
Refer to the Canadian Journal of Civil Engineering (Jan. 2013) investigation of the use of variable speed limits to control freeway traffic congestion, Exercise 4.9. Recall that the study site was an
Refer to the Engineering Computations: International Journal for Computer-Aided Engineering and Software (Vol. 30, No. 2, 2013) study of the properties of granular media (e.g., sand, rice, ball
Refer to the IEEE Transactions on Information Forensics and Security (March 2013) study of wireless identify theft using cloned credit or debit cards, Exercise 3.44. A cloning detection method was
Consider 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 die and let Y be the outcome on the second die.a. Find the joint
A software program is designed to perform two tasks, A and B. Let X represent the number of IF-THEN statements in the code for task A and let Y represent the number of IF-THEN statements in the code
Let my(t) be the moment generating function of a continuous random variable Y. If a and b are constants, show thata.b.c. my+a(l) = E[e(y+a] = e"m,(1) ealm,(1)
To aid engineers seeking to predict the efficiency of a solar-powered device, Olseth and Skartveit developed a model for daily insolation Y at sea-level locations within the temperate storm belt. To
The continuous random variable Y has a probability distribution given bya. Find the value of c that makes f(y) a probability density.b. Find F(y).c. Compute P(Y > 2.5). (cye if y > 0 f(y) elsewhere
Let c be a constant and consider the density function for the random variable Y:a. Find the value of c.b. Find the cumulative distribution function F(y).c. Compute F(2.6).d. Show that and F(0) = 0
In finding and correcting errors in a software code (debugging) and determining the code’s reliability, computer software experts have noted the importance of the distribution of the time until the
Refer to the Journal of the Acoustical Society of America (Feb. 1986) study of auditory nerve response rates in cats, discussed in Exercise 4.107. A key question addressed by the research is whether
The life length Y (in years) of a memory chip in a laptop computer is a Weibull random variable with probability densitya. What are the values of α and β?b. Compute the mean and variance of Y.c.
The percentage Y of impurities per batch in a certain chemical product is a beta random variable with probability densitya. What are the values of α and β?b. Compute the mean and variance of Y.c. A
Show that for a Poisson random variable Y, σ2 = λ. ( Use the result of Exercise 4.83 and Theorem 4.4.)Data from Exercise 4.83 Elevator passenger arrivals. A study of the arrival process of people
Verify that the mean of a beta density with parameters α and β is given by μ = α/(α + β).
Prove Theorem 5.1. THEOREM 5.1 Let c be a constant, let Y be a continuous random variable, and let g1(Y), g2(Y), ..., gx(Y) be k functions of Y. Then, E(c) = c E(cY) = cE(Y) E[g (Y) + g2(Y) + + g(Y)]
Prove Theorem 5.2. THEOREM 5.2 Let Y be a continuous random variable with E(Y) = µ. Then 7 = E[CY – p1 = E(Y?) – p? %3D
Assume that Y is uniformly distributed over the interval 0 ≤ Y ≤ 1. Show that, for a ≥ 0 b, b ≥ 0, and (a + b) ≤ 1,P(a < Y < a + b) = b
Engineers studied the life cycle cost of a coal-fired power plant in the Journal of Quality in Maintenance Engineering (Vol. 19, 2013). The analysis used the Weibull distribution to model the
For each of the following exercises, find μ and σ2. Then compute P(μ – 2σ < Y < μ + 2σ) and compare to the Empirical Rule.a. Exercise 5.1b. Exercise 5.2c. Exercise 5.3d.
Refer to Example 5.15 (p. 197) and the life length Y of a drill bit. Recall that Y has a Weibull distribution with α = 2 and β = 100.a. Calculate the values of f(y) for y = 2, 5, 8, 11, 14 and 17.
Japanese electrical engineers have developed a sophisticated radar system called the moving target detector (MTD), designed to reject ground clutter, rain clutter, birds, and other interference. The
Based on extensive testing, a manufacturer of washing machines believes that the distribution of the time Y (in years) until a major repair is required has a Weibull distribution with α = 2 and β =
The length of time (in months after maintenance) until failure of a bank’s surveillance television equipment has a Weibull distribution with α = 2 and β = 60. If the bank wants the probability of
Show that the Weibull distribution with α = 2 and β > 0 is new better than used (NBU).
Show that for the Weibull distribution, a +
Show that for the Weibull distribution,Then use the relationshipto show that EÇY²) = B2/«T( a + 2
Sedimentary rocks are a mixture of minerals and pores. Engineers employ well (or borehole) logging to make a detailed record of the mineral composition of the rock penetrated by a borehole. One
At the 2006 International Conference on Biomedical Engineering, researchers presented an unsupervised learning technique for segmenting skin images. The method employed various forms of a beta
In the Journal of Statistical Computation and Simulation (Vol. 67, 2000), researchers presented a numerical procedure for estimating the parameters of a beta distribution. The method involves a
A block cipher is an encryption algorithm that transforms a fixed-length block of unencrypted text data (called plaintext) into a block of encrypted text data (called ciphertext) of the same length
An investigation into pollution control expenditures of industrial firms found that the annual percentage of plant capacity shutdown attributable to environmental and safety regulation has an
The proportion Y of a data processing company’s yearly hardware repair budget allocated to repair its laser color printer has an approximate beta distribution with parameters α = 2 and β = 9.a.
Suppose the proportion of Internet start-up firms that make a profit during their first year of operation possesses a relative frequency distribution that can be approximated by the beta density with
An important property of certain products that are in powder or granular form is their particle size distribution. For example, refractory cements are adversely affected by too high a proportion of
A continuous random variable Y has a beta distribution with probability densityFind the value of c that will make f(y) a density function. Scy'(1 - y) if 0 s ys 1 f(y) elsewhere
Show that if Y has a beta density with α = 1and β = 1, then Y is uniformly distributed over the interval 0 ≤ Y ≤ 1.
Show that the beta distribution with α = 2 and β = 1 is new better than used (NBU).
Use the moment generating function m(t) of the normal density to find μ1'and μ2' . Then use these results to show that a normal random variable has mean μ and variance σ2.
Verify that the moment generating function of a chi-square random variable with n degrees of freedom is m(t) = (1 - 2t)-v/2
Consider a continuous random variable Y with densitya. Find the moment generating function m(t) of Y.b. Use the result of part a to find the mean and variance of Y. Se if y < 0 Ry) lo elsewhere
Verify that the moment generating function of a uniform random variable on the interval a ≤ Y ≤ b is elb - eta m(t) t(b – a)
Ecological Applications (May 1995) published a study on the development of forests following wildfires in the Pacific Northwest. One variable of interest to the researcher was tree diameter at breast
A manufacturing company has developed a fuel-efficient machine that combines pressure washing with steam cleaning. It is designed to deliver 7 gallons of cleaner per minute at 1,000 pounds per square
The problem of passenger congestion prompted a large international airport to install a monorail connecting its main terminal to the three concourses, A, B, and C. The engineers designed the monorail
A pacemaker is made up of several biomedical components that must be of a high quality for the pacemaker to work. It is vitally important for manufacturers of pacemakers to use parts that meet
An infestation of a certain species of caterpillar, the spruce budworm, can cause extensive damage to the timberlands of the northern United States. It is known that an outbreak of this type of
In an article published in the European Journal of Operational Research, the vehicle-dispatching decisions of an airport based taxi service were investigated. In modeling the system, the authors
The importance of modeling machine downtime correctly in simulation studies was discussed in Industrial Engineering (Aug. 1990). The paper presented simulation results for a single-machine tool
Suppose that the fraction of defective modems shipped by a data-communications vendor has an approximate beta distribution with α = 5 and β = 21.a. Find the mean and variance of the fraction of
W. Nelson (Journal of Quality Technology, July 1985) suggests that the Weibull distribution usually provides a better representation for the life length of a product than the exponential
Digital elevation models (DEM) are now used to estimate elevations and slopes of remote regions. In Arctic, Antarctic, and Alpine Research (May 2004), geographers analyzed reading errors from maps
The Harris Corporation data on voltage readings at two locations, Exercise 2.72, are reproduced at the bottom of the page. Determine whether the voltage readings at each location are approximately
A team of soil scientists investigated the water retention properties of soil cores sampled from an uncropped field consisting of silt loam (Soil Science, Jan. 1995). At a pressure of .1 megapascal
Refer to the data on percentage iron in 66 bulk specimens of Chilean lumpy iron ore, Exercise 2.79. The data are saved in the LUMPYORE file. Assess whether the data are approximately normal.Data From
Suppose we are counting events that occur according to a Poisson distribution, such as the number of automobile accidents at a left-turn lane. If it is known that exactly one such event has occurred
Refer to the Journal of Earthquake Engineering study of the time Y (in years) between major earthquakes occurring in the Iranian Plateau, Exercise 5.7. Recall that Y has the following density
Refer to Exercises 4.2 and 4.13. Each home with a dust mite level that exceeds 2mg/g will spend $2,000 for an allergen air purification system. Find the mean and variance of the total amount spent by
Use Theorem 4.4 to calculate the variance of the probability distribution in Exercise 4.4. Interpret the result.Data from Exercise 4.4 The data on the nearshore bar condition for six beach hot spots
Use Theorem 4.4 to calculate the variance of the probability distribution in Exercise 4.12. Verify that your result agrees with Exercise 4.12.Data from Exercise 4.12According to an August, 2011
Refer to Exercise 4.11, where Y is the number of firing pins tested in a sample of five selected from a large lot. Suppose the cost of inspecting a single pin is $300 if the pin is defective and $100
Refer to the Annals of the Entomological Society of America (Jan. 2005) study of the life cycle of a South American delphacid species, Exercise 4.3 (p. 138). Recall that entomological engineers have
The Environmental Protection Agency (EPA) has established national ambient air quality standards in an effort to control air pollution. Currently, the EPA limit on ozone levels in air is 12 parts per
The nuclear industry has made a concerted effort to significantly reduce the number of unplanned rapid emergency shutdowns of a nuclear reactor— called scrams. A decade ago, the mean annual number
Show that for a Poisson random variable Y, a. 0 s p(y) s 1 b Σρ)-1 y=0 C. E(Y) = A? + A [Hint: First derive the result E[ Y(Y – 1)] = x? from the fact that ELY(Y – 1)] = Ey(y - 1)- y! y=0
Derive the moment generating function of the Poisson random variable Y. WriteThen note that the quantity being summed is a Poisson probability with parameter λet.] m(t) = E(e") = E nd'e %3! y=0 y!
Use the result of Exercise 4.86 to derive the mean and variance of the Poisson distribution.
Researchers at the University of Rochester studied the friction that occurs in the paper feeding process of a photocopier (Journal of Engineering for Industry, May 1993). The coefficient of friction
Timber beams are widely used in home construction. When the load (measured in pounds) per unit length has a constant value over part of a beam the load is said to be uniformly distributed over that
Statistical software packages, such as SAS and MINITAB, are capable of generating random numbers from a uniform distribution. For example, the SAS function RANUNI uses a prime modulus 231 – 1
Assume that the random variable Y is uniformly distributed over the interval a ≤ Y ≤ b. Verify the following:a.b. a + b (b – a)? and o? 12
Show that the uniform distribution is new better than used (NBU) over the interval (0, 1).
The Transactions of the ASME (June 2004) presented a model for predicting daily natural gas consumption in urban areas. A key component of the model is the distribution of daily temperatures in the
Seismic ground noise describes the persistent vibration of the ground due to surface waves generated from traffic, heavy machinery, winds, ocean waves, and earthquakes. A group of civil engineers
As part of a risk assessment, quality engineers monitored the corrosion rate (millimeters per year) of a wind turbine system susceptible to corrosion. (Journal of Quality in Maintenance Engineering,
The alkalinity level of water specimens collected from the Han River in Seoul, Korea, has a mean of 50 milligrams per liter and a standard deviation of 3.2 milligrams per liter. (Environmental
Paleomagnetic studies of Canadian volcanic rock known as the Carmacks Group have recently been completed. The studies revealed that the northward displacement of the rock units has an approximately
Let Y be a normal random variable with mean m and variance σ2. Show thathas mean 0 and variance 1. Y - u Z =
Refer to the 2010 CSI Computer Crime and Security Survey, Exercise 2.13. Recall that the percentage of monetary losses attributable to malicious actions by individuals within the organization (i.e.,
An evaluation of the habitats of endangered salmon species was performed in Conservation Ecology (December 2003). The researchers identified 734 sites (habitats) for Chinook, coho, or steelhead
Foresters periodically “cruise” a forest to determine the size (usually measured as the diameter at breast height) of a certain species of trees. The breast height diameters (in meters) for a
Refer to the National Highway Traffic Safety Administration (NHTSA) crash test data for new cars. In Exercise 5.38, you assumed that the driver’s head injury rating is approximately normally
Refer to the data on sanitation scores for 186 cruise ships, first presented in Exercise 2.19. The data are saved in the SHIPSANIT file. Assess whether the sanitation scores are approximately
Refer to the Minerals Engineering (Vol. 46-47, 2013) study of the impact of calcium and gypsum on the flotation properties of silica in water, Exercise 2.23. Recall that 50 solutions of deionized
The optimal scheduling of preventative maintenance tests of some (but not all) of n independently operating components was developed in Reliability Engineering and System Safety (Jan., 2006). The
Refer to the Chance (Summer, 2007) article on phishing attacks at a company, Exercise 2.24. Recall that phishing describes an attempt to extract personal/financial information
Researchers have discovered that the maximum flood level (in millions of cubic feet per second) over a 4-year period for the Susquehanna River at Harrisburg, Pennsylvania, follows approximately a
Effective maintenance of equipment depends on the ability to accurately forecast the demand for spare parts. The Journal of Quality in Maintenance Engineering (Vol. 18, 2012) developed a statistical
Reaction to tear gas. The length of time Y (in minutes) required to generate a human reaction to tear gas formula A has a gamma distribution with α = 2 and β = 2. The distribution for formula B is
Show that the variance of a gamma distribution with parameters α and β is αβ2.

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