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introduction to probability statistics
Probability And Statistics For Engineering And The Sciences 8th Edition Jay L Devore, Roger Ellsbury - Solutions
The paper “Microwave Observations of Daily Antarctic Sea-Ice Edge Expansion and Contribution Rates” (IEEE Geosci. and Remote Sensing Letters, 2006: 54–58) states that “The distribution of the daily sea-ice advance/retreat from each sensor is similar and is approximately double
Let the time between two successive arrivals at the drive-up window of a local bank. If X has an exponential distribution with (which is identical to a standard gamma distribution with ), compute the following:a. The expected time between two successive arrivalsb. The standard deviation of the time
There is no nice formula for the standard normal cdf , but several good approximations have been published in articles. The following is from “Approximations for Hand Calculators Using Small Integer Coefficients” (Mathematics of Computation, 1977: 214–222). For 0 , z # 5.5,(z)8F 8C The
a. Show that if X has a normal distribution with parameters and , then (a linear function of X)also has a normal distribution. What are the parameters of the distribution of Y [i.e., E(Y) and V(Y)]? [Hint:Write the cdf of Y, , as an integral involving the pdf of X, and then differentiate with
Let X have a binomial distribution with parameters and p. Calculate each of the following probabilities using the normal approximation (with the continuity correction) for the cases , .6, and .8 and compare to the exact probabilities calculated from Appendix Table A.1.a.b.c.
Chebyshev’s inequality, (see Exercise 44, Chapter 3), is valid for continuous as well as discrete distributions. It states that for any number satisfying ,(see Exercise 44 in Chapter 3 for an interpretation). Obtain this probability in the case of a normal distribution for , 2, and 3, and compare
In response to concerns about nutritional contents of fast foods, McDonald’s has announced that it will use a new cooking oil for its french fries that will decrease substantially trans fatty acid levels and increase the amount of more beneficial polyunsaturated fat. The company claims that 97
Suppose Appendix Table A.3 contained only for Explain how you could still compute a.b.Is it necessary to tabulate for z negative? What property of the standard normal curve justifies your answer?
The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 70 and standard deviation 3.(Rockwell hardness
The article “Monte Carlo Simulation—Tool for Better Understanding of LRFD” (J. of Structural Engr., 1993:1586–1599) suggests that yield strength (ksi) for A36 grade steel is normally distributed with and .a. What is the probability that yield strength is at most 40?Greater than 60?b. What
a. If a normal distribution has and , what is the 91st percentile of the distribution?b. What is the 6th percentile of the distribution?c. The width of a line etched on an integrated circuit chip is normally distributed with mean and standard deviation .140. What width value separates the widest
Spray drift is a constant concern for pesticide applicators and agricultural producers. The inverse relationship between droplet size and drift potential is well known.The paper “Effects of 2,4-D Formulation and Quinclorac on Spray Droplet Size and Deposition” (Weed Technology, 2005:
The article “Reliability of Domestic-Waste Biofilm Reactors” (J. of Envir. Engr., 1995: 785–790) suggests that substrate concentration of influent to a reactor is normally distributed with and .a. What is the probability that the concentration exceeds .25?b. What is the probability that the
Mopeds (small motorcycles with an engine capacity below) are very popular in Europe because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the Maximum Speed of Automatic Transmission Mopeds in Periodic Motor Vehicle Inspections” (J. of Automobile Engr.,
Suppose the force acting on a column that helps to support a building is a normally distributed random variable X with mean value 15.0 kips and standard deviation 1.25 kips.Compute the following probabilities by standardizing and then using Table A.3.a. b.c. d.e.
Determine for the following:a. b.c.
In each case, determine the value of the constant c that makes the probability statement correct.a. b.c. d.e.
Let Z be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate.a. b.c. d.e. f.g. h.i. j.
When a dart is thrown at a circular target, consider the location of the landing point relative to the bull’s eye. Let X be the angle in degrees measured from the horizontal, and assume that X is uniformly distributed on [0, 360]. Define Y to be the transformed variable , so Y is the angle
Let X be the total medical expenses (in 1000s of dollars)incurred by a particular individual during a given year.Y 5 aX 1 b 1.8m| 1 32 m| Y 5 1.8X 1 32 8F 8C E(Xn)k 5 2 V(X) 5 ku2(k 2 1)22(k 2 2)21 k . 2 k 5 1 k . 1 f(x; k, u) 5 u k # uk xk11 x $ u 0 x , u 8F 8F 5 1.88C 1 32 28C 1208C Although X is
Let X be the temperature in at which a certain chemical reaction takes place, and let Y be the temperature in (so).a. If the median of the X distribution is , show that is the median of the Y distribution.b. How is the 90th percentile of the Y distribution related to the 90th percentile of the X
Let X have the Pareto pdf introduced in Exercise 10.a. If , compute E(X).b. What can you say about E(X) if ?c. If , show that .d. If , what can you say about V(X)?e. What conditions on k are necessary to ensure that is finite?
If the temperature at which a certain compound melts is a random variable with mean value and standard deviation , what are the mean temperature and standard deviation measured in ? [Hint: .]
Consider the pdf for total waiting time Y for two buses introduced in Exercise 8.a. Compute and sketch the cdf of Y. [Hint: Consider separately and in computing F(y). A graph of the pdf should be helpful.]b. Obtain an expression for the (100p)th percentile. [Hint:Consider separately and .]c.
Let X be a continuous rv with cdf[This type of cdf is suggested in the article “Variability in Measured Bedload-Transport Rates” (Water Resources Bull., 1985: 39–48) as a model for a certain hydrologic variable.] What isa. ?b. ?c. The pdf of X?
Let X denote the voltage at the output of a microphone, and suppose that X has a uniform distribution on the interval from to 1.The voltage is processed by a “hard limiter”with cutoff values and .5, so the limiter output is a random variable Y related to X by if if, and if .a. What is ?b.
Let X have a uniform distribution on the interval [A, B].a. Obtain an expression for the (100p)th percentile.b. Compute E(X), V(X), and .c. For n, a positive integer, compute .
Answer parts (a)–(f) of Exercise 15 with lecture time past the hour given in Exercise 5.
Let X denote the amount of space occupied by an article placed in a 1- packing container. The pdf of X isa. Graph the pdf. Then obtain the cdf of X and graph it.b. What is [i.e., F(.5)]?c. Using the cdf from (a), what is ? What is ?d. What is the 75th percentile of the distribution?e. Compute E(X)
The cdf for X ( measurement error) of Exercise 3 isa. Compute .b. Compute .c. Compute .d. Verify that f(x) is as given in Exercise 3 by obtaining.e. Verify that .
Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdf is Use the cdf to obtain the following:a.b.c.d. The median checkout duration [solvee. to obtain the density function f(x)f. E(X)g. V(X) and h. If the borrower is charged an amount when checkout
A family of pdf’s that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, k and , both , and the pdf isa. Sketch the graph of .b. Verify that the total area under the graph equals 1.c. If the rv X
In commuting to work, a professor must first get on a bus near her house and then transfer to a second bus. If the waiting time (in minutes) at each stop has a uniform distribution with and , then it can be shown that the total waiting time Y has the pdf A 5 0 B 5 5 a 1 2 25 , a , a 1 2 , 35 mA 5
The time X (min) for a lab assistant to prepare the equipment for a certain experiment is believed to have a uniform distribution with and .a. Determine the pdf of X and sketch the corresponding density curve.b. What is the probability that preparation time exceeds 33 min?c. What is the probability
A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X isa. Find the value of k and draw the corresponding
Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. The article“Blade Fatigue Life Assessment with Application to VAWTS” (J. of Solar Energy Engr., 1982: 107–111) proposes the Rayleigh distribution, with pdf as a model for the X
The error involved in making a certain measurement is a continuous rv X with pdfa. Sketch the graph of f(x).b. Compute .c. Compute .d. Compute . P(X , 2.5 or X . .5)P(21 , X , 1)P(X . 0)f(x) 5 e.09375(4 2 x2) 22 # x # 2 0 otherwise P(k , X , k 1 4)25 , k , k 1 4 , 5 P(22 # X # 3)P(22.5 , X ,
Suppose the reaction temperature X (in ) in a certain chemical process has a uniform distribution with and .a. Compute .b. Compute .c. Compute .d. For k satisfying , compute.
The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following density function:a. Graph the pdf and verify that the total area under the density curve is indeed 1.b. Calculate . How does this probability compare to ?c. Calculate and also .
Consider a communication source that transmits packets containing digitized speech. After each transmission, the receiver sends a message indicating whether the transmission was successful or unsuccessful. If a transmission is unsuccessful, the packet is re-sent. Suppose a voice packet can be
Consider a collection A1, . . . , Ak of mutually exclusive and exhaustive events, and a random variable X whose distribution depends on which of the Ai’s occurs (e.g., a commuter might select one of three possible routes from home to work, with X representing the commute time). Let denote the
The simple Poisson process of Section 3.6 is characterized by a constant rate at which events occur per unit time. A generalization of this is to suppose that the probability of exactly one event occurring in the interval is. It can then be shown that the number of events occurring during an
If X is a hypergeometric rv, show directly from the definition that (consider only the case ).[Hint: Factor nM/N out of the sum for E(X), and show that the terms inside the sum are of the form, where .]
A computer disk storage device has ten concentric tracks, numbered 1, 2, . . . , 10 from outermost to innermost, and a single access arm. Let that any particular request for data will take the arm to track. Assume that the tracks accessed in successive seeks are independent. Let X 5 the number of
The mode of a discrete random variable X with pmf p(x) is that value x* for which p(x) is largest (the most probable x value).a. Let . By considering the ratio, show that b(x; n, p) increases with x as long as . Conclude that the mode x* is the integer satisfying .b. Show that if X has a Poisson
There are two Certified Public Accountants in a particular office who prepare tax returns for clients. Suppose that for a particular type of complex form, the number of errors made by the first preparer has a Poisson distribution with mean value m1, the number of errors made by the second preparer
The generalized negative binomial pmf is given by Let X, the number of plants of a certain species found in a particular region, have this distribution with and. What is ? What is the probability that at least one plant is found?
A test for the presence of a certain disease has probability.20 of giving a false-positive reading (indicating that an individual has the disease when this is not the case) and 1 2 P(X , 20)P(20 # X) P(20 # X) 5 1 2 p 2 q 5 P(draw)p 5 P(S) q 5 P(F)p(x) 5 P(X 5 x)x 5 10, 11,c, 19 X 5 the number P(F)
Individuals A and B begin to play a sequence of chess games. Let , and suppose that outcomes of successive games are independent with and (they never draw). They will play until one of them wins ten games. Let of games played (with possible values 10, 11, . . . , 19).a. For , obtain an expression
A newsstand has ordered five copies of a certain issue of a photography magazine. Let of individuals who come in to purchase this magazine. If X has a Poisson distribution with parameter , what is the expected number of copies that are sold?
Grasshoppers are distributed at random in a large field according to a Poisson process with parameter per square yard. How large should the radius R of a circular sampling region be taken so that the probability of finding at least one in the region equals .99?
A reservation service employs five information operators who receive requests for information independently of one another, each according to a Poisson process with rate per minute.a. What is the probability that during a given 1-min period, the first operator receives no requests?b. What is the
A trial has just resulted in a hung jury because eight members of the jury were in favor of a guilty verdict and the other four were for acquittal. If the jurors leave the jury room in random order and each of the first four leaving the room is accosted by a reporter in quest of an interview, what
Forty percent of seeds from maize (modern-day corn) ears carry single spikelets, and the other 60% carry paired spikelets. A seed with single spikelets will produce an ea with single spikelets 29% of the time, whereas a seed with paired spikelets will produce an ear with single spikelets 26% of the
A plan for an executive travelers’ club has been developed by an airline on the premise that 10% of its current customers would qualify for membership.a. Assuming the validity of this premise, among 25 randomly selected current customers, what is the probability that between 2 and 6 (inclusive)
The purchaser of a power-generating unit requires c consecutive successful start-ups before the unit will be accepted. Assume that the outcomes of individual start-ups are independent of one another. Let p denote the probability that any particular start-up is successful. The random variable of
Consider a disease whose presence can be identified by carrying out a blood test. Let p denote the probability that a randomly selected individual has the disease. Suppose n individuals are independently selected for testing. One way to proceed is to carry out a separate test on each of the n blood
An educational consulting firm is trying to decide whether high school students who have never before used a handheld calculator can solve a certain type of problem more easily with a calculator that uses reverse Polish logic or one that does not use this logic. A sample of 25 students is selected
Of the people passing through an airport metal detector,.5% activate it; let among a randomly selected group of 500 who activate the detector.a. What is the (approximate) pmf of X?b. Compute .c. Compute .
Of all customers purchasing automatic garage-door openers, 75% purchase a chain-driven model. Let number among the next 15 purchasers who select the chain-driven model.a. What is the pmf of X?b. Compute .c. Compute .d. Compute m and s2.e. If the store currently has in stock 10 chain-driven models
The negative binomial rv X was defined as the number of F’s preceding the rth S. Let of trials necessary to obtain the rth S. In the same manner in which the pmf of X was derived, derive the pmf of Y.
After shuffling a deck of 52 cards, a dealer deals out 5.Let of suits represented in the five-card hand.a. Show that the pmf of X is[Hint: , (only spades and hearts with at least one of each suit), and .]b. Compute m, s2 , and s.5 4P(2 spades ¨ one of each
Consider a deck consisting of seven cards, marked 1, 2, . . . ,7.Three of these cards are selected at random. Define an rv W by of the resulting numbers, and compute the pmf of W. Then compute m and s2. [Hint: Consider outcomes as unordered, so that (1, 3, 7) and (3, 1, 7) are not different
a. In a Poisson process, what has to happen in both the time interval (0, t) and the interval so that no events occur in the entire interval ? Use this and Assumptions 1–3 to write a relationship between and P0(t).b. Use the result of part (a) to write an expression for the difference . Then
Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate per hour. Suppose that with probability .5 an arriving vehicle will have no equipment violations.a 5 10 1 sq mile 5 640 acres X 5 the number E(X) 5 ma. What is the probability that exactly ten
Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parametera, the expected number of trees per acre, equal to 80.a. What is the probability that in a certain quarter-acre plot, there will be at most 16 trees?b. If the forest covers 85,000 acres, what
Let X have a Poisson distribution with parameter m. Show that directly from the definition of expected value.[Hint: The first term in the sum equals 0, and then x can be canceled. Now factor out m and show that what is left sums to 1.]
The article “Reliability-Based Service-Life Assessment of Aging Concrete Structures” (J. Structural Engr., 1993:1600–1621) suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is .5 yeaa. How
The number of requests for assistance received by a towing service is a Poisson process with rate per hour.a. Compute the probability that exactly ten requests are received during a particular 2-hour period.b. If the operators of the towing service take a 30-min break for lunch, what is the
Suppose small aircraft arrive at a certain airport according to a Poisson process with rate per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter .a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At
Suppose that only .10% of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 10,000 computers.m 5 .2 m 5 20 P(8 # X)P(4 # X # 8)P(X # 4) P(X , 4)m 5 4 P(5 # X # 8) P(5 , X , 8)P(X # 8) P(X 5 8) P(9 # X)m 5 5a. What are the expected value and
Consider writing onto a computer disk and then sending it through a certifier that counts the number of missing pulses.Suppose this number X has a Poisson distribution with parameter . (Suggested in “Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution,” J. Quality
Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter(suggested in the article “Dynamic Ride Sharing: Theory and Practice,” J. of Transp. Engr., 1997: 308–312). What is the
Let X be the number of material anomalies occurring in a particular region of an aircraft gas-turbine disk. The article“Methodology for Probabilistic Life Prediction of MultipleAnomaly Materials” (Amer. Inst. of Aeronautics and Astronautics J., 2006: 787–793) proposes a Poisson distribution
Let X, the number of flaws on the surface of a randomly selected boiler of a certain type, have a Poisson distribution with parameter . Use Appendix Table A.2 to compute the following probabilities:a.b. c.d. e.
According to the article “Characterizing the Severity and Risk of Drought in the Poudre River, Colorado” (J. of Water Res. Planning and Mgmnt., 2005: 383–393), the drought X 5 the total number X 5 the number P(B) 5 P(G) 5 .5 length Y is the number of consecutive time intervals in which the
Three brothers and their wives decide to have children until each family has two female children. What is the pmf of of male children born to the brothers?What is E(X), and how does it compare to the expected number of male children born to each brother?
A family decides to have children until it has three children of the same gender. Assuming , what is the pmf of of children in the family?
Suppose that . A couple wishes to have exactly two female children in their family. They will have children until this condition is fulfilled.a. What is the probability that the family has x male children?b. What is the probability that the family has four children?c. What is the probability that
A second-stage smog alert has been called in a certain area of Los Angeles County in which there are 50 industrial firms. An inspector will visit 10 randomly selected firms to check for violations of regulations.a. If 15 of the firms are actually violating at least one regulation, what is the pmf
In the first part of the tournament, the 20 are randomly divided into 10 east–west pairs and 10 north–south pairs.a. What is the probability that x of the top 10 pairs end up playing east–west?b. What is the probability that all of the top five pairs end up playing the same direction?c. If
A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis.P(X # 5)P(X # 4)P(X 5 5)P(X 5 2) P(X # 2) P(X $ 2)X 5 the numbera. What is the pmf of the number of granite
An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30, decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first 15 graded projects.a. What
Each of 12 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators
An electronics store has received a shipment of 20 table radios that have connections for an iPod or iPhone. Twelve of these have two slots (so they can accommodate both devices), and the other eight have a single slot. Suppose that six of the 20 radios are randomly selected to be stored under a
Refer to Chebyshev’s inequality given in Exercise 44.Calculate for and when, and compare to the corresponding upper bound. Repeat for X , Bin(20, .75).
An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reservation. From previous records, 20% of all those making reservations do not appear for the trip. Answer the following
Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make independent choices, with , , and.a. Among the next 100 customers, what are the mean and variance of the number who pay with a debit card?Explain your reasoning.b. Answer part
Show that when X is a binomial random variable. [Hint: First express E(X) as a sum with lower limit. Then factor out np, let so that the sum is from to , and show that the sum equals 1.] y 5 0 y 5 n 2 1 x 5 1 y 5 x 2 1 E(X) 5 np(n 2 x)B(x; n, 1 2 p) 5 1 2 B(n 2 x 2 1; n, p)b(x; n, 1 2 p) 5 b(n 2 x;
a. Show that .b. Show that .[Hint: At most x S’s is equivalent to at least F’s.]c. What do parts (a) and (b) imply about the necessity of including values of p greater than .5 in Appendix Table A.1?
a. For fixed n, are there values of for which? Explain why this is so.b. For what value of p is V(X) maximized? [Hint: Either graph V(X) as a function of p or else take a derivative.]
A toll bridge charges $1.00 for passenger cars and $2.50 for other vehicles. Suppose that during daytime hours, 60%of all vehicles are passenger cars. If 25 vehicles cross the bridge during a particular daytime period, what is the resulting expected toll revenue? [Hint: Let of passenger cars; then
An ordinance requiring that a smoke detector be installed in all previously constructed houses has been in effect in a particular city for 1 year. The fire department is concerned that many houses remain without detectors. Let proportion of such houses having detectors, and suppose that a random
A very large batch of components has arrived at a distributor. The batch can be characterized as acceptable only if the proportion of defective components is at most .10. The distributor decides to randomly select 10 components and to accept the batch only if the number of defective components in
The College Board reports that 2% of the 2 million high school students who take the SAT each year receive special accommodations because of documented disabilities (Los Angeles Times, July 16, 2002). Consider a random sample of 25 students who have recently taken the test.a. What is the
Suppose that 30% of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other 70% want a used copy. Consider randomly selecting 25 purchasers.a. What are the mean value and standard deviation of the number who want a new copy of the book?b. What
A company that produces fine crystal knows from experience that 10% of its goblets have cosmetic flaws and must be classified as “seconds.”a. Among six randomly selected goblets, how likely is it that only one is a second?P(1 # X # 4)P(X $ 5)P(X # 2)n 5 25 X , Bin(25, .05)X 5 the number P(2 , X
If , show that . a # X # b a # E(X) # b P(u X 2 m u
A result called Chebyshev’s inequality states that for any probability distribution of an rv X and any number k that is at least 1, . In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/k2.a. What is the value of the upper bound for ? ?? ?
a. Draw a line graph of the pmf of X in Exercise 35.Then determine the pmf of and draw its line graph. From these two pictures, what can you say about V(X) and?b. Use the proposition involving to establish a general relationship between V(X) and .
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