Questions and Answers of Applied Statistics and Probability

Construct a histogram for the semiconductor speed data in Exercise 6-42. Comment on the shape of the histogram. Does it convey the same information as the stem-andleaf display?Data from 6-42 680 669
Construct a histogram for the spot weld shear strength data in Exercise 6-39. Comment on the shape of the histogram. Does it convey the same information as the stem-and-leaf display?Data from 6-39
Construct a histogram for the female student height data in Exercise 6-38.Data from 6-38 62 64 61 67 65 68 61 65 60 65 64 63 59 65 67 62 66 68 67 66 65 67 65 68 64 66 68 69 69 65 69 65 67 67 65 63 4
Construct a histogram for the energy consumption data in Exercise 6-37.Data from 6-37 Billions of Kilowatt-Hours Afghanistan 1.04 Australia 200.66 Bangladesh 16.20 6.88 Burma 1671.23 China 38.43 Hong
Construct histograms with 8 and 16 bins for the data in Exercise 6-32. Compare the histograms. Do both histograms display similar information?Data from 6-32 37.8 33.6 34.2 32.6 33.8 35.8 34.7 34.6
Construct histograms with 8 and 16 bins for the data in Exercise 6-31. Compare the histograms. Do both histograms display similar information?Data from 6-31 1115 865 1015 885 1594 1000 1416 1501 1310
Construct frequency distributions and histograms with 8 bins and 16 bins for the motor fuel octane data in Exercise 6-30. Compare the histograms. Do both histograms display similar information?Data
Construct a frequency distribution and histogram for the yield data in Exercise 6-33.Data from 6-33 94.1 86.1 95.3 84.9 88.8 84.6 94.4 84.1 78.3 93.2 90.4 94.1 86.4 83.6 96.1 83.7 89.6 98.0 90.6 89.1
Construct a frequency distribution and histogram for the cotton content data in Exercise 6-32.Data from 6-32 88.5 98.8 89.6 92.2 92.7 88.4 87.5 90.9 94.7 89.9 88.3 90.4 83.4 87.9 92.6 87.8 84.3 90.4
Table 6E.10 shows unemployment data for the United States that are seasonally adjusted. Construct a time series plot of these data and comment on any features (source: U.S. Bureau of Labor Web site,
A sample of six resistors yielded the following resistances (ohms): x1 = 45, x2 = 38, x3 = 47, x4 = 41, x5 = 35, and x6 = 43.(a) Compute the sample variance and sample standard deviation.(b) Subtract
The total net electricity consumption of the United States by year from 1980 to 2007 (in billion kilowatt-hours) is in Table 6E.12. Net consumption excludes the energy consumed by the generating
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.
In Exercise 5-31, 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
Three electron emitters produce electron beams with changing kinetic energies that are uniformly distributed in the ranges [3,7], [2,5], and [4,10]. Let Y denote the total kinetic energy produced by
A U-shaped component is to be formed from the three parts A, B, and C. See Fig. 5-18. The length of A is normally distributed with a mean of 10 mm and a standard deviation of 0.1 mm. The thickness of
For the Transaction Processing Performance Council’s benchmark in Exercise 5-10, let X, Y, and Z denote the average number of selects, updates, and inserts operations required for each type of
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
Determine the covariance and correlation for the joint probability distribution shown in Fig. 5-10(a) and described in Example 5-10.In figure 5-10 fy(y) 4 0.034 0.1275 0.0085 0.17 3 0.1875 0.050
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
An article in Health Economics [€œEstimation of the Transition Matrix of a Discrete-Time Markov Chain€ (2002, Vol.11, pp. 33€“42)] considered the changes in CD4 white
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)]
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 the minor and major axes (in micrometers),
Determine the value of c that makes fXYZ (x, y, z) = c a joint probability density function over the region x > 0, y > 0, z > 0, and x + y + z < 1.Determine the following:(a) P(X <0.5,
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
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
The blade and the bearings are important parts of a lathe. The lathe can operate only when both of them work properly. The lifetime of the blade is exponentially distributed with the mean three
Two methods of measuring surface smoothness are used to evaluate a paper product. The measurements are recorded as deviations from the nominal surface smoothness in coded units. The joint probability
Determine the value of c that makes the function f (x, y) = ce−2x−3y, a joint probability density function over the range 0 < x and x < y.Determine the following:(a) P(X <1, Y
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 0 < y < x.Determine the following:(a) P(X <1, Y
Determine the value of c that makes the function f (x, y) = ce− 2x− 3y a joint probability density function over the range 0 < x and 0 < y < x.Determine the following:(a) P(X <1, Y
In the transmission of digital information, the probability that a bit has high, moderate, or low distortion is 0.01, 0.04, and 0.95, respectively. Suppose that three bits are transmitted and that
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,
Determine the value of c such that the function f (x, y) = cxy for 0 < x < 3 and 0 < y < 3 satisfies the properties of a joint probability density function.Determine the following:(a) P(X
For the Transaction Processing Performance Council’s benchmark in Exercise 5-10, let X, Y , and Z denote the average number of selects, updates, and inserts operations required for each type of
An article in the Journal of Database Management [“Experimental Study of a Self-Tuning Algorithm for DBMS Buffer Pools” (2005, Vol. 16, pp. 1–20)] provided the workload used in the TPC-C OLTP
An engineering statistics class has 40 students; 60% are electrical engineering majors, 10% are industrial engineering majors, and 30% are mechanical engineering majors. A sample of four students is
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 <1.5) (d) P(X = 1 or Z =
Show that the following function satisfies the properties of a joint probability mass function.Determine the following:(a) P(X <0.5, Y <1.5) (b) P(X <0.5)(c) P(Y <1.5) (d) P(X
Determine the value of c that makes the function f (x, y) = c(x + y) a joint probability mass function over the nine points with x = 1, 2, 3 and y = 1, 2, 3. Determine the following:(a) P(X = 1, Y
Show that the following function satisfies the properties of a joint probability mass function.Determine the following:(a) P(X <2.5, Y <3) (b) P(X <2.5)(c) P(Y <3) (d) P(X >1.8, Y >
The two-parameter exponential distribution uses a different range for the random variable X, namely, 0 ≤ γ ≤ x for a constant γ (and this equals the usual exponential distribution in the
Determine the mean and variance of a beta random variable. Use the result that the probability density function integrates to 1. That is, Γ (α)Γ(β) T(a+ß) = [,x(1 – x)8-1 for a > 0, ß > 0.
Lack of Memory Property. Show that for an exponential random variable X, P(X < t1 + t2 u X > t1) = P(X < t2).
Consider the endothermic reactions given below. Let A denote the event that a reaction's final temperature is 271 K or less. Let B denote the event that the heat absorbed is above target.Determine
Consider the hospital emergency room data given below. Let A denote the event that a visit is to hospital 4, and let B denote the event that a visit results in LWBS (at any hospital). Use the
Consider the bar code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (five bars
The analysis of results from a leaf transmutation experiment (turning a leaf into a petal) is summarized by type of transformation completed:(a) If a leaf completes the color transformation, what is
A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Assume
Consider the well failure data given below. Let A denote the event that the geological formation has more than 1000 wells, and let B denote the event that a well failed. Determine the following
An acticle in Biometrics [“Integrative Analysis of Transcriptomic and Proteomic Data of Desulfovibrio Vulgaris: A Nonlinear Model to Predict Abundance of Undetected Proteins” (2009)] reported
A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. Suppose that an operating room needs to schedule three knee, four hip, and five shoulder
The article [“Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early Rheumatoid Arthritis,” Arthritis & Rheumatism (2005, Vol. 52, pp.
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
A computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9).
Redundant array of inexpensive disks (RAID) is a technology that uses multiple hard drives to increase the speed of data transfer and provide instant data backup. Suppose that the probability of any
A player of a video game is confronted with a series of four opponents and an 80% probability of defeating each opponent. Assume that the results from opponents are independent (and that when the
Six tissues are extracted from an ivy plant infested by spider mites. The plant in infested in 20% of its area. Each tissue is chosen from a randomly selected area on the ivy plant.(a) What is the
In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Because acids and bases are usually colorless (as are the water and salt
Consider the endothermic reactions given below. Let A denote the event that a reaction's final temperature is 271 K or less. Let B denote the event that the heat absorbed is above
Consider the well failure data given below. Let A denote the event that the geological formation has more than 1000 wells, and let B denote the event that a well failed. Are these events independent?
The article €œTerm Efficacy of Ribavirin Plus Interferon Alfa in the Treatment of Chronic Hepatitis C,€ [Gastroenterology (1996, Vol. 111, no. 5, pp. 1307€“1312)],
The article [“Clinical and Radiographic Outcomes of Four Different Treatment Strategies in Patients with Early Rheumatoid Arthritis,” Arthritis & Rheumatism (2005, Vol. 52, pp.
A steel plate contains 20 bolts. Assume that 5 bolts are not torqued to the proper limit. 4 bolts are selected at random, without replacement, to be checked for torque. If an operator checks a bolt,
In an acid-base titration, the milliliters of base that are needed to reach equivalence are measured to the nearest milliliter between 0.1 and 0.15 liters (inclusive).
The number of mutations in a nucleotide sequence of length 40,000 in a DNA strand after exposure to radiation is measured. Each nucleotide may be mutated.
Consider the hospital patients in Example 2-8. Two patients are selected randomly, with replacement, from the total patients at Hospital. What is the probability mass function of the number of
Orders arrive at a Web site according to a Poisson process with a mean of 12 per hour. Determine the following:(a) Probability of no orders in five minutes.(b) Probability of 3 or more orders in five
A total of 12 cells are replicated. Freshly synthesized DNA cannot be replicated again until mitosis is completed. Two control mechanisms have been identified—one positive and one negative—that
The distribution of X is approximated with a triangular probability density function f (x) = 0.025x − 0.0375 for 30 < x < 50 and f (x) = −0.025x + 0.0875 for 50 < x < 70.Determine the
The waiting time for service at a hospital emergency department (in hours) follows a distribution with probability density function f (x) = 0.5exp(−0.5x) for 0 < x. Determine the following:(a)
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
The probability density function of the weight of packages delivered by a post office is f (x) = 70 / (69x2) for 1 < x < 70 pounds.(a) Determine the mean and variance of weight.(b) If the
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
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.
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
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
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
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.
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
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
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
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
An article in Atmospheric Chemistry and Physics “Relationship Between Particulate Matter and Childhood Asthma—Basis of a Future Warning System for Central Phoenix” (2012, Vol. 12, pp.
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
An article in International Journal of Electrical Power & Energy Systems [“Stochastic Optimal Load Flow Using a Combined Quasi–Newton and Conjugate Gradient Technique” (1989, Vol.11(2), pp.
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
An article in Electric Power Systems Research [€œModeling Real-Time Balancing Power Demands in Wind Power Systems Using Stochastic Differential Equations€ (2010, Vol. 80(8), pp.
Derive the convergence results used to obtain a Poisson distribution as the limit of a binomial distribution.
Suppose that 50 sites on a patient might contain lesions. A biopsy selects 8 sites randomly (without replacement). What is the minimum number of sites with lesions so that the probability of at least
Saguaro cacti are large cacti indigenous to the southwestern United States and Mexico. Assume that the number of saguaro cacti in a region follows a Poisson distribution with a mean of 280 per square
Messages that arrive at a service center for an information systems manufacturer have been classified on the basis of the number of keywords (used to help route messages) and the type of message,
Traffic flow is traditionally modeled as a Poisson distribution. A traffic engineer monitors the traffic flowing through an intersection with an average of six cars per minute. To set the timing of a
An automated egg carton loader has a 1% probability of cracking an egg, and a customer will complain if more than one egg per dozen is cracked. Assume that each egg load is an independent event.(a)
Inclusions are defects in poured metal caused by contaminants. The number of (large) inclusions in cast iron follows a Poisson distribution with a mean of 2.5 per cubic millimeter. Determine the
The article “An Association Between Fine Particles and Asthma Emergency Department Visits for Children in Seattle” [Environmental Health Perspectives June, 1999 107(6)] used Poisson models for
Cabs pass your workplace according to a Poisson process with a mean of five cabs per hour. Suppose that you exit the workplace at 6:00 p.m. Determine the following:(a) Probability that you wait more
The number of views of a page on a Web site follows a Poisson distribution with a mean of 1.5 per minute.(a) What is the probability of no views in a minute?(b) What is the probability of two or
Astronomers treat the number of stars in a given volume of space as a Poisson random variable. The density in the Milky Way Galaxy in the vicinity of our solar system is one star per 16 cubic
Verify that the following functions are probability mass functions, and determine the requested probabilities.1.(a) P(X ‰¤ 2)(b) P(X > ˆ’2)(c) P(ˆ’1‰¤ X

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