Answer: Suppose that 30% of all students who have to buy a

Problem 16E Some parts of California are particularly earthquake-prone. Suppose that in one metropolitan area, 25% of all homeowners are insured against earthquake damage. Four homeowners are to be selected at random; let X denote the number among the four who have earthquake insurance. a. Find the probability distribution of X. [Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS, with probability (.25)(.75)(.25)(.25) and associated X value 3. There are15 other outcomes.] b. Draw the corresponding probability histogram. c. What is the most likely value for X? d. What is the probability that at least two of the four selected have earthquake insurance?

Problem 17E A new battery’s voltage may be acceptable (A) or unacceptable (U). A certain flashlight requires two batteries, so batteries will be independently selected and tested until two acceptable ones have been found. Suppose that 90% of all batteries have acceptable voltages. Let Y denote the number of batteries that must be tested. a. What is p(2), that isP ( Y = 2), ? b. What is p(3)? [Hint: There are two different outcomes that result inY = 3 .] c. To have Y=5, what must be true of the fifth battery selected? List the four outcomes for which Y= 5 and then determine p(5). d. Use the pattern in your answers for parts (a)–(c) to obtain a general formula for p(y).

Problem 18E Two fair six-sided dice are tossed independently. Let M = the maximum of the two tosses (so M (1,5) =5, M(3,3,) = 3, etc.). a. What is the pmf of M? [Hint: First determine p(1), then p(2), and so on.] b. Determine the cdf of M and graph it.

Problem 38E

Problem 39E A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 5-lb batches. Let X = the number of batches ordered by a randomly chosen customer, and suppose that X has pmf Compute E(X) and V(X). Then compute the expected number of pounds left after the next customer’s order is shipped and the variance of the number of pounds left. [Hint: The number of pounds left is a linear function of X.]

Problem 40E a. Draw a line graph of the pmf of X in Exercise 35. Then determine the pmf of –X and draw its line graph. From these two pictures, what can you say about V(X) and V(- X)? b. Use the proposition involving V( aX = b) to establish a general relationship between V(X) and V( -X). Reference exercise -35 A small market orders copies of a certain magazine for its magazine rack each week. Let X = demand for the magazine, with pmf Suppose the store owner actually pays $2.00 for each copy of the magazine and the price to customers is $4.00. If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine? [Hint: For both three and four copies ordered, express net revenue as a function of demand X, and then compute the expected revenue.]

Problem 59E 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 p = the true proportion of such houses having detectors, and suppose that a random sample of 25 homes is inspected. If the sample strongly indicates that fewer than 80% of all houses have a detector, the fire department will campaign for a mandatory inspection program. Because of the costliness of the program, the department prefers not to call for such inspections unless sample evidence strongly argues for their necessity. Let X denote the number of homes with detectors among the 25 sampled. Consider rejecting the claim that p ? .8 if x ? 15. a. What is the probability that the claim is rejected when the actual value of p is .8? b. What is the probability of not rejecting the claim when p = .7? when p = .6? c. How do the “error probabilities” of parts (a) and (b) change if the value 15 in the decision rule is replaced by 14?

Problem 60E 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 X = the number of passenger cars; then the toll revenue h(X) is a linear function of X.]

A student who is trying to write a paper for a course has a choice of two topics, A and B. If topic A is chosen, the student will order two books through interlibrary loan, whereas if topic B is chosen, the student will order four books. The student believes that a good paper necessitates receiving and using at least half the books ordered for either topic chosen. If the probability that a book ordered through interlibrary loan actually arrives in time is .9 and books arrive independently of one another, which topic should the student choose to maximize the probability of writing a good paper? What if the arrival probability is only .5 instead of .9?

Problem 80E 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 Multiple- Anomaly Materials” (Amer. Inst. of Aeronautics and Astronautics J., 2006: 787–793) proposes a Poisson distribution for X. Suppose that ? = 4. a. compute both P (X ? 4) and P ( X < 4). b. Compute P( 4 ? X ? 8) c. compute P (8 ? X). d. What is the probability that the number of anomalies exceeds its mean value by no more than one standard deviation?

Problem 81E 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 ? =20 (suggested in the article “Dynamic Ride Sharing: Theory and Practice,” J. of Transp. Engr., 1997: 308–312). What is the probability that the number of drivers will a. Be at most 10? b. Exceed 20? c. Be between 10 and 20, inclusive? Be strictly between 10 and 20? d. Be within 2 standard deviations of the mean value?

Problem 82E 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 ? = .2. (Suggested in “Average Sample Number for Semi-Curtailed Sampling Using the Poisson Distribution,” J. Quality Technology, 1983: 126–129.) a. What is the probability that a disk has exactly one missing pulse? b. What is the probability that a disk has at least two missing pulses? c. If two disks are independently selected, what is the probability that neither contains a missing pulse?

Problem 102E 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 and allowed to practice on both calculators. Then each student is asked to work one problem on the reverse Polish calculator and a similar problem on the other. Let p =P(S), where S indicates that a student worked the problem more quickly using reverse Polish logic than without, and let X = number of S’s. a. If p = .5 , what is P( 7 ? X ? 18)? b. If p = .8, what is P( 7 ? X ? 18)? c. If the claim that p = .5 is to be rejected when either x ? 7 or x ? 8 , what is the probability of rejecting the claim when it is actually correct? d. If the decision to reject the claim is made as in part (c), what is the probability that the claim is not rejected when p = .6? When p = .8? e. What decision rule would you choose for rejecting the claim p = .5 if you wanted the probability in part (c) to be at most .01?

Problem 103E 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 samples. A potentially more economical approach, group testing, was introduced during World War II to identify syphilitic men among army inductees. First, take a part of each blood sample, combine these specimens, and carry out a single test. If no one has the disease, the result will be negative, and only the one test is required. If at least one individual is diseased, the test on the combined sample will yield a positive result, in which case the n individual tests are then carried out. If p = .1 and n = 3, what is the expected number of tests using this procedure? What is the expected number when n = 5? [The article “Random Multiple-Access Communication and Group Testing” (IEEE Trans. on Commun., 1984: 769–774) applied these ideas to a communication system in which the dichotomy was active/idle user rather than diseased/nondiseased.]

Problem 104E Let p1 denote the probability that any particular code symbol is erroneously transmitted through a communication system. Assume that on different symbols, errors occur independently of one another. Suppose also that with probability p2 an erroneous symbol is corrected upon receipt. Let X denote the number of correct symbols in a message block consisting of n symbols (after the correction process has ended). What is the probability distribution of X?

A library subscribes to two different weekly news magazines, each of which is supposed to arrive in Wednesday’s mail. In actuality, each one may arrive on Wednesday, Thursday, Friday, or Saturday. Suppose the two arrive independently of one another, and for each one P(Wed.) = .3, P(Thurs.) = .4, P(Fri.) = .2, and P(Sat.) = .1. Let Y = the number of days beyond Wednesday that it takes for both magazines to arrive (so possible Y values are 0, 1, 2, or 3). Compute the pmf of Y. [Hint: There are 16 possible outcomes; Y(W,W) = 0, Y(F,Th) = 2, and so on.]

Problem 20E Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number of people who arrive late for the seminar. a. Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.] b. Obtain the cumulative distribution function of X, and use it to calculate P(2? X ? 6).

Problem 21E Suppose that you read through this year’s issues of the New York Times and record each number that appears in a news article—the income of a CEO, the number of cases of wine produced by a winery, the total charitable contribution of a politician during the previous tax year, the age of a celebrity, and so on. Now focus on the leading digit of each number, which could be 1, 2, . . . , 8, or 9. Your first thought might be that the leading digit X of a randomly selected number would be equally likely to be one of the nine possibilities (a discrete uniform distribution). However, much empirical evidence as well as some theoretical arguments suggest an alternative probability distribution called Benford’s law: a. Without computing individual probabilities from this formula, show that it specifies a legitimate pmf. b. Now compute the individual probabilities and compare to the corresponding discrete uniform distribution. c. Obtain the cdf of X. d. Using the cdf, what is the probability that the leading digit is at most 3? At least 5? [Note: Benford’s law is the basis for some auditing procedures used to detect fraud in financial reporting—for example, by the Internal Revenue Service.]

Problem 42E Suppose E(X) = 5 E[X(X - 1)] = 27.5 What is a. E(X2)? [Hint: E[X(X - 1)] = E[X2 - X] = E(X2) - E(X)]? b. V(X)? c. The general relationship among the quantities E(X), E[X(X - 1)], and V(X)?

Problem 41E Use the definition in Expression (3.13) to prove that V(aX + b) = ?2. ?2x [ Hint: With h(X) = aX + b, E[h(X)] = aµ = b where µ = E(X).] Reference Expression (3.13 The variance of h(X) is the expected value of the squared difference between h(X) and its expected value: When h(X) = aX + b, a linear function, Substituting this into (3.13) gives a simple relationship between V[h(X)] and V(X):

Problem 43E Write a general rule for E(X - c) where c is a constant. What happens when you let c - µ , the expected value of X?

Problem 62E a. For fixed n, are there values of p (0 ? p ? 1) for which V(X) = 0? 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.]

Problem 63E a. Show that b(x; n, 1 - p) = b(n - x; n, p). b. Show that B(x; n, 1 – p) = 1 – B ( n – x – 1; n, p). [Hint: At most x S’s is equivalent to at least ( n – x) 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?

Problem 64E Show that E(X) = np when X is a binomial random variable. [Hint: First express E(X) as a sum with lower limit x = 1.Then factor out np, let y = x - 1 so that the sum is from y = 0 to y = n - 1, and show that the sum equals 1.]

Problem 83E An article in the Los Angeles Times (Dec. 3, 1993) reports that 1 in 200 people carry the defective gene that causes inherited colon cancer. In a sample of 1000 individuals, what is the approximate distribution of the number who carry this gene? Use this distribution to calculate the approximate probability that a. Between 5 and 8 (inclusive) carry the gene. b. At least 8 carry the gene.

Problem 84E The Centers for Disease Control and Prevention reported in 2012 that 1 in 88 American children had been diagnosed with an autism spectrum disorder (ASD). a. If a random sample of 200 American children is selected, what are the expected value and standard deviation of the number who have been diagnosed with ASD? b. Referring back to (a), calculate the approximate probability that at least 2 children in the sample have been diagnosed with ASD? c. If the sample size is 352, what is the approximate probability that fewer than 5 of the selected children have been diagnosed with ASD?

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate \(\alpha=8\) per hour, so that the number of arrivals during a time period of thours is a Poisson rv with parameter \(\mu=8 \mathrm{t}\). a. What is the probability that exactly 6 small aircraft arrive during a 1-hour period? At least 6? At least 10? b. What are the expected value and standard deviation of the number of small aircraft that arrive during a 90-min period? c. What is the probability that at least 20 small aircraft arrive during a 2.5-hour period? That at most 10 arrive during this period?

Problem 105E 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 interest is X = the number of start-ups that must be made prior to acceptance. Give the pmf of X for the case c= 2. If p =.9, what is P(X ? 8)? [Hint: for x ? 5, express p(x) “recursively” in terms of the pmf evaluated at the smaller values x - 3, x – 4,……,2.] (This problem was suggested by the article “Evaluation of a Start-Up Demonstration Test,” J. Quality Technology, 1983: 103–106.)

Problem 106E 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) qualify for membership? b. Again assuming the validity of the premise, what are the expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of 100 current customers? c. Let X denote the number in a random sample of 25 current customers who qualify for membership. Consider rejecting the company’s premise in favor of the claim that p > .10 if x ? 7. What is the probability that the company’s premise is rejected when it is actually valid? d. Refer to the decision rule introduced in part (c). What is the probability that the company’s premise is not rejected even though p = .20 (i.e., 20% qualify)?

Problem 107E 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 ear with single spikelets 29% of the time, whereas a seed with paired spikelets will produce an ear with single spikelets 26% of the time. Consider randomly selecting ten seeds. a. What is the probability that exactly five of these seeds carry a single spikelet and produce an ear with a single spikelet? b. What is the probability that exactly five of the ears produced by these seeds have single spikelets? What is the probability that at most five ears have single spikelets?

Problem 1E A concrete beam may fail either by shear (S) or flexure (F). Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let X= the number of beams among the three selected that failed by shear. List each outcome in the sample space along with the associated value of X.

Problem 2E Give three examples of Bernoulli rv’s (other than those in the text).

Problem 3E Using the experiment in Example 3.3, define two more random variables and list the possible values of each. Reference example 3.3 Example 2.3 described an experiment in which the number of pumps in use at each of two six-pump gas stations was determined. Define rv’s X, Y, and U by X= the total number of pumps in use at the two stations Y =the difference between the number of pumps in use at station 1 and the number in use at station 2 U= the maximum of the numbers of pumps in use at the two stations If this experiment is performeds= (2,3) and results, then X((2,3)) =2+3 =5 , so we say that the observed value of X was x= 5 . Similarly, the observed value of Y would be y =2-3 = -1, and the observed value of U would be u = max (2,3) -3. Each of the random variables of Examples 3.1–3.3 can assume only a finite number of possible values. This need not be the case.

Problem 22E Refer to Exercise 13, and calculate and graph the cdf F(x). Then use it to calculate the probabilities of the events given in parts (a)–(d) of that problem. Reference Exercise 13 p>A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. Calculate the probability of each of the following events. a. {at most three lines are in use} b. {fewer than three lines are in use} c. {at least three lines are in use} d. {between two and five lines, inclusive, are in use} e. {between two and four lines, inclusive, are not in use} f. {at least four lines are not in use}

Problem 23E

Problem 24E An insurance company offers its policyholders a number of different premium payment options. For a randomly selected policyholder, let X = the number of months between successive payments. The cdf of X is as follows: a. What is the pmf of X? b. Using just the cdf, compute P(3 ? X ? 6) and P( 4 ? X).

Problem 44E 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, P( | X - µ | k ?) ? 1/k2 . 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 k = 2? K + 3? K = 4? K= 5? K = 10? b. Compute µ and ? for the distribution of Exercise 13. Then evaluate P(|X - µ| ? k?) for the values of k given in part (a). What does this suggest about the upper bound relative to the corresponding probability? c. Let X have possible values -1, 0, and 1, with probabilities 1/18, 8/9 and 1/8 , respectively. What is P(|X - µ|? 3?), and how does it compare to the corresponding bound? d. Give a distribution for which P(|X - µ|? 5?) = .04. Reference exercise -13 A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. Calculate the probability of each of the following events. a. {at most three lines are in use} b. {fewer than three lines are in use} c. {at least three lines are in use} d. {between two and five lines, inclusive, are in use} e. {between two and four lines, inclusive, are not in use} f. {at least four lines are not in use}

If a ? X ? b, show that a ? E(X) ? b.

Problem 46E Compute the following binomial probabilities directly from the formula for b(x; n, p): a. b(3; 8, .35) b. b(5; 8, .6) c.P(3 ? X ? 5) when n= 7 and p = .6 d. P(1? X) when n = 9 p = .1

Problem 65E 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 P(A) = .5, P(B) = .2 and P(C) = .3. 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 (a) for the number among the 100 who don’t pay with cash.

Problem 66E 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 questions, assuming independence wherever appropriate. a. If six reservations are made, what is the probability that at least one individual with a reservation cannot be accommodated on the trip? b. If six reservations are made, what is the expected number of available places when the limousine departs? c. Suppose the probability distribution of the number of reservations made is given in the accompanying table. Let X denote the number of passengers on a randomly selected trip. Obtain the probability mass function of X.

Problem 67E Refer to Chebyshev’s inequality given in Exercise 44. Calculate P( | X - ? | ? k?) for k = 2 and k = 3 when X ? Bin (20,. 5) , and compare to the corresponding upper bound. Repeat for X ? Bin (20,. 75) Reference exercise 44 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 P(|X – ?| ? k?) ? 1/k2 . 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 k = 2? K = 3? k = 4 ? k+ 5? K+ 10? b. Compute ? and ? for the distribution of Exercise 13. Then evaluate P(| X - ? | ? k? ) for the values of k given in part (a). What does this suggest about the upper bound relative to the corresponding probability? c. Let X have possible values -1, 0, and 1, with probabilities 1/18 , 8/9 and 1/ 18, respectively. What is P( |X - ?| ? 3?), and how does it compare to the corresponding bound? d. Give a distribution for which P( |X - ?| ? 5?) = . 04

Problem 86E Organisms are present in ballast water discharged from a ship according to a Poisson process with a concentration of 10 organisms/m3 [the article "Counting at Low Concentrations: • The Statistical Challenges of Verifying Ballast Water Discharge Standards" (Ecological Applications, 2013: 339-351) considers using the Poisson process for this purpose]. a. What is the probability that one cubic meter of discharge contains at least 8 organisms? b. What is the probability that the number of organisms in 1.5 m3 of discharge exceeds its mean value by more than one standard deviation? c. For what amount of discharge would the probability of containing at least 1 organism be .999?

Problem 87E The number of requests for assistance received by a towing service is a Poisson process with ? = 4 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 probability that they do not miss any calls for assistance? c. How many calls would you expect during their break?

Problem 88E In proof testing of circuit boards, the probability that any particular diode will fail is .01. Suppose a circuit board contains 200 diodes. a. How many diodes would you expect to fail, and what is the standard deviation of the number that are expected to fail? b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board? c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (A board works properly only if all its diodes work.)

Problem 108E 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 is the pmf of X = the number of jurors favoring acquittal among those interviewed? How many of those favoring acquittal do you expect to be

Problem 109E A reservation service employs five information operators who receive requests for information independently of one another, each according to a Poisson process with rate ? = 2 per minute. a. What is the probability that during a given 1-min period, the first operator receives no requests? b. What is the probability that during a given 1-min period, exactly four of the five operators receive no requests? c. Write an expression for the probability that during a given 1-min period, all of the operators receive exactly the same number of requests.

Problem 110E Grasshoppers are distributed at random in a large field according to a Poisson process with parameter ? = 2 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?

Problem 4E Let X = the number of nonzero digits in a randomly selected zip code. What are the possible values of X? Give three possible outcomes and their associated X values.

Problem 5E If the sample space will have an infinite set of possible values? If yes, say why. If no, give an example.

Problem 6E Starting at a fixed time, each car entering an intersection is observed to see whether it turns left (L), right (R), or goes straight ahead (A). The experiment terminates as soon as a car is observed to turn left. Let X= the number of cars observed. What are possible X values? List five outcomes and their associated X values.

Problem 26E Alvie Singer lives at 0 in the accompanying diagram and has four friends who live at A, B, C, and D. One day Alvie decides to go visiting, so he tosses a fair coin twice to decide which of the four to visit. Once at a friend’s house, he will either return home or else proceed to one of the two adjacent houses (such as 0, A, or C when at B), with each of the three possibilities having probability . In this way, Alvie continues to visit friends until he returns home. a. Let X = the number of times that Alvie visits a friend. Derive the pmf of X. b. Let Y= the number of straight-line segments that Alvie traverses (including those leading to and from 0). What is the pmf of Y? c. Suppose that female friends live at A and C and male friends at B and D. If Z = the number f visits to female friends, what is the pmf of Z?

Problem 27E After all students have left the classroom, a statistics professor notices that four copies of the text were left under desks. At the beginning of the next lecture, the professor distributes the four books in a completely random fashion to each of the four students (1, 2, 3, and 4) who claim to have left books. One possible outcome is that 1 receives 2’s book, 2 receives 4’s book, 3 receives his or her own book, and 4 receives 1’s book. This outcome can be abbreviated as (2, 4, 3, 1). a. List the other 23 possible outcomes. b. Let X denote the number of students who receive their own book. Determine the pmf of X.

Problem 28E Show that the cdf F(x) is a nondecreasing function; that is x1 < x2, implies that F(x1) ? F(x2). Under what condition will F(x1) = F(x2) ?

Problem 47E The article "Should You Report That Fender-Bender?" (Consumer Reports, Sept. 2013: 15) reported that 7 in 10 auto accidents involve a single vehicle (the article recommended always reporting to the insurance company an accident involving multiple vehicles). Suppose 15 accidents are randomly selected. Use Appendix Table A.1 to answer each of the following questions. a. What is the probability that at most 4 involve a single vehicle? b. What is the probability that exactly 4 involve a single vehicle? c. What is the probability that exactly 6 involve multiple vehicles? d. What is the probability that between 2 and 4, inclusive, involve a single vehicle? e. What is the probability that at least 2 involve a single vehicle? f. What is the probability that exactly 4 involve a single vehicle and the other 11 involve multiple vehicles?

Problem 48E

Problem 49E 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? b. Among six randomly selected goblets, what is the probability that at least two are seconds? c. If goblets are examined one by one, what is the probability that at most five must be selected to find four that are not seconds?

Eighteen individuals are scheduled to take a driving test at a particular DMV office on a certain day, eight of whom will be taking the test for the first time. Suppose that six of these individuals are randomly assigned to a particular examiner, and let \(X\) be the number among the six who are taking the test for the first time. a. What kind of a distribution does \(X\) have (name and values of all parameters)? b. Compute \(P(X=2), P(X \leq 2)\), and \(P(X \geq 2)\). c. Calculate the mean value and standard deviation of \(X\).

Problem 69E 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 are examined in random order, let X be the number among the first 6 examined that have a defective compressor. a. Calculate P(X = 4) and P(X ? 4) b. Determine the probability that X exceeds its mean value by more than 1 standard deviation. c. Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 15 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X ?5) than to use the hypergeometric pmf.

Problem 70E 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 is the probability that exactly 10 of these are from the second section? b. What is the probability that at least 10 of these are from the second section? c. What is the probability that at least 10 of these are from the same section? d. What are the mean value and standard deviation of the number among these 15 that are from the second section? e. What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section?

Problem 89E 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 year. a. How many loads can be expected to occur during a 2- year period? b. What is the probability that more than five loads occur during a 2-year period? c. How long must a time period be so that the probability of no loads occurring during that period is at most .1?

Problem 90E Let X have a Poisson distribution with parameter ?. Show that E( X ) = ? 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.]

Problem 91E Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter a, 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 is the expected number of trees in the forest? c. Suppose you select a point in the forest and construct a circle of radius .1 mile. Let X = the number of trees within that circular region. What is the pmf of X? [Hint: 1sq mile = 640 acres.]

Problem 111E A newsstand has ordered five copies of a certain issue of a photography magazine. Let X = the number of individuals who come in to purchase this magazine. If X has a Poisson distribution with parameter ? = 4, what is the expected number of copies that are sold?

Problem 112E Individuals A and B begin to play a sequence of chess games. Let S = { A wins a game }, and suppose that outcomes of successive games are independent with P(S) – p and P(F) – 1 – p )they never draw). They will play until one of them wins ten games. Let X = the number of games played (with possible values 10, 11,……..,19). a. For x = 10, 11, …..19, obtain an expression for p(x) – P(X = x). b. If a draw is possible, with p = P(S), q = P(F), 1- p – q = P(draw), what are the possible values os X? What is P(20 ? X) = 1- P(X < 20).]

Problem 113E 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 probability .10 of giving a false-negative result. Suppose that ten individuals are tested, five of whom have the disease and five of whom do not. Let X = the number of positive readings that result. a. Does X have a binomial distribution? Explain your reasoning. b. What is the probability that exactly three of the ten test results are positive?

Problem 7E For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete. , a. X = the number of unbroken eggs in a randomly chosen standard egg carton b. Y = the number of students on a class list for a particular course who are absent on the first day of classes c. U = the number of times a duffer has to swing at a golf ball before hitting it d. X = the length of a randomly selected rattlesnake e. Z = the sales tax percentage for a randomly selected amazon.com purchase f. Y = the pH of a randomly chosen soil sample g. X = the tension (psi) at which a randomly selected tennis racket has been strung h. X = the total number of times three tennis players must spin their rackets to obtain something other than UUU or DDD (to determine which two play next)

Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y, and state which Y value is associated with each one.

Problem 9E An individual named Claudius is located at the point 0 in the accompanying diagram. Using an appropriate randomization device (such as a tetrahedral die, one having four sides), Claudius first moves to one of the four locations B1, B2, B3, B4. Once at one of these locations, another randomization device is used to decide whether Claudius next returns to 0 or next visits one of the other two adjacent points. This process then continues; after each move, another move to one of the (new) adjacent points is determined by tossing an appropriate die or coin. a. Let X = the number of moves that Claudius makes before first returning to 0. What are possible values of X? Is X discrete or continuous? b. If moves are allowed also along the diagonal paths connecting 0 to A1, A2, A3, and A4, respectively, answer the questions in part (a).

The pmf of the amount of memory \(X\) (GB) in a purchased flash drive was given in Example x 1 2 4 8 16 p(x) .05 .10 .35 .40 .10 Compute the following: a. \(E(X)\) b. \(V(X)\) directly from the definition c. The standard deviation of \(X\) d. \(V(X)\) using the shortcut formula

Problem 30E An individual who has automobile insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the last 3 years. The pmf of Y is a. Compute E(Y). b. Suppose an individual with Y violations incurs a surcharge of $100Y2. Calculate the expected amount of the surcharge.

Problem 31E Refer to Exercise 12 and calculate V(Y) and sY. Then determine the probability that Y is within 1 standard deviation of its mean value. Reference exercise -12 Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table. a. What is the probability that the flight will accommodate all ticketed passengers who show up? b. What is the probability that not all ticketed passengers who show up can be accommodated? c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. What is the probability that a. At most 6 of the calls involve a fax message? b. Exactly 6 of the calls involve a fax message? c. At least 6 of the calls involve a fax message? d. More than 6 of the calls involve a fax message?

Problem 51E Refer to the previous exercise. a. What is the expected number of calls among the 25 that involve a fax message? b. What is the standard deviation of the number among the 25 calls that involve a fax message? c. What is the probability that the number of calls among the 25 that involve a fax transmission exceeds the expected number by more than 2 standard deviations?

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 is the probability that the number who want new copies is more than two standard deviations away from the mean value? c. The bookstore has 15 new copies and 15 used copies in stock. If 25 people come in one by one to purchase this text, what is the probability that all 25 will get the type of book they want from current stock? [Hint: Let X = the number who want a new copy. For what values of X will all 25 get what they want?] d. Suppose that new copies cost $100 and used copies cost $70. Assume the bookstore currently has 50 new copies and 50 used copies. What is the expected value of total revenue from the sale of the next 25 copies purchased? Be sure to indicate what rule of expected value you are using. [Hint: Let h(X) = the revenue when X of the 25 purchasers want new copies. Express this as a linear function.]

Problem 71E 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. a. What is the pmf of the number of granite specimens selected for analysis? b. What is the probability that all specimens of one of the two types of rock are selected for analysis? c. What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?

Problem 72E A personnel director interviewing 11 senior engineers for four job openings has scheduled six interviews for the first day and five for the second day of interviewing. Assume that the candidates are interviewed in random order. a. What is the probability that x of the top four candidates are interviewed on the first day? b. How many of the top four candidates can be expected to be interviewed on the first day?

Problem 73E Twenty pairs of individuals playing in a bridge tournament have been seeded 1, . . . , 20. 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 there are 2n pairs, what is the pmf of X = the number among the top n pairs who end up playing east–west? What are E(X) and V(X)?

Problem 92E Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with rate ? = 10 per hour. Suppose that with probability .5 an arriving vehicle will have no equipment violations. a. What is the probability that exactly ten arrive during the hour and all ten have no violations? b. For any fixed y ? 10, what is the probability that y arrive during the hour, of which ten have no violations? c. What is the probability that ten “no-violation” cars arrive during the next hour? [Hint: Sum the probabilities in part (b) from y =10 to ?.]

Problem 94E Consider a deck consisting of seven cards, marked 1, 2, . . . , 7. Three of these cards are selected at random. Define an rv W by W = the sum 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 outcomes. Then there are 35 outcomes, and they can be listed. (This type of rv actually arises in connection with a statistical procedure called Wilcoxon’s rank-sum test, in which there is an x sample and a y sample and W is the sum of the ranks of the x’s in the combined sample; see Section 15.2.)

Problem 95E After shuffling a deck of 52 cards, a dealer deals out 5. Let X = the number of suits represented in the five-card hand. a. Show that the pmf of X is [Hint: p(1) = 4P( all are spades),p(2) =6P( only spades and hearts with at least one of each suit), and p(4) = 4P(2 spades ? one of each other suit). b.Compute ?, ?2 , and ?.

Problem 114E 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 p = .3 and r =2.5. What is P( X = 4)? What is the probability that at least one plant is found?

Problem 115E 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 ?1, the number of errors made by the second preparer has a Poisson distribution with mean value ?2, and that each CPA prepares the same number of forms of this type. Then if a form of this type is randomly selected, the function gives the pmf of X = the number of errors on the selected form. a. Verify that p(x; ?1, ?2) is in fact a legitimate pmf ( ? 0and sums to 1). b. What is the expected number of errors on the selected form? c. What is the variance of the number of errors on the selected form? d. How does the pmf change if the first CPA prepares 60% of all such forms and the second prepares 40%?

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 X ~Bin(n, p). By considering the ratio b(x + 1; n, p) / b(x; n, p), show that b(x; n, p) increases with x as long as x < np – ( 1 – p). Conclude that the mode x* is the integer satisfying \((n + 1)p - 1 \leq x^* \leq (n + 1) p\). b. Show that if X has a Poisson distribution with parameter ?, the mode is the largest integer less than ?. If \(\mu\) is an integer, show that both \(\mu -1\) and \(\mu\) are modes.

Problem 10E The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for a. T = the total number of pumps in use b. X = the difference between the numbers in use at stations 1 and 2 c. U = the maximum number of pumps in use at either station d. Z= the number of stations having exactly two pumps in use

Let X be the number of students who show up for a professor's office hour on a particular day. Suppose that the pmf of X is p(0) = .20, p(1) = .25, p(2) = .30, p(3) = .15, and p(4) = .10. a. Draw the corresponding probability histogram. b. What is the probability that at least two students show up? More than two students show up? c. What is the probability that between one and three students, inclusive, show up? d. What is the probability that the professor shows up?

Problem 12E Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table. a. What is the probability that the flight will accommodate all ticketed passengers who show up? b. What is the probability that not all ticketed passengers who show up can be accommodated? c. If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

Problem 32E

Problem 33E Let X be a Bernoulli rv with pmf as in Example 3.18. a. Compute E(X2). b. Show that V(X) = p(1 – p). c. Compute E(X79). Reference example 3.18 Let X = 1 if a randomly selected vehicle passes an emissions test and X = 0 otherwise. Then X is a Bernoulli rv with pmf p(1) = p and p(0) = 1 - p , from which E(X) = 0. P(0) + 1. P(1) = 0(1 – p) + 1(p) = p. That is, the expected value of X is just the probability that X takes on the value 1. If we conceptualize a population consisting of 0s in proportion and 1 – p in proportion p, then the population average is µ = p.

Problem 34E Suppose that the number of plants of a particular type found in a rectangular sampling region (called a quadrat by ecologists) in a certain geographic area is an rv X with pmf Is E(X) finite? Justify your answer (this is another distribution that statisticians would call heavy-tailed).

Problem 53E Exercise 30 (Section 3.3) gave the pmf of Y, the number of traffic citations for a randomly selected individual insured by a particular company. What is the probability that among 15 randomly chosen such individuals a. At least 10 have no citations? b. Fewer than half have at least one citation? c. The number that have at least one citation is between 5 and 10, inclusive?* Reference exercise 30 An individual who has automobile insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the last 3 years. The pmf of Y is a. Compute E(Y). b. Suppose an individual with Y violations incurs a surcharge of $100Y2. Calculate the expected amount of the surcharge.

Problem 54E A particular type of tennis racket comes in a midsize version and an oversize version. Sixty percent of all customers at a certain store want the oversize version. a. Among ten randomly selected customers who want this type of racket, what is the probability that at least six want the oversize version? b. Among ten randomly selected customers, what is the probability that the number who want the oversize version is within 1 standard deviation of the mean value? c. The store currently has seven rackets of each version. What is the probability that all of the next ten customers who want this racket can get the version they want from current stock?

Problem 55E Twenty percent of all telephones of a certain type are submitted for service while under warranty. Of these, 60% can be repaired, whereas the other 40% must be replaced with new units. If a company purchases ten of these telephones, what is the probability that exactly two will end up being replaced under warranty?

The probability that a randomly selected box of a certain type of cereal has a particular prize is .2. Suppose you purchase box after until you have obtained two of these prizes. a. What is the probability that you purchase x boxes that do not have the desired prize? b. What is the probability that you purchase four boxes? c. What is the probability that you purchase at most four boxes? d. How many boxes without the desired prize do you expect to purchase? How many boxes do you expect to purchase?

Problem 76E A family decides to have children until it has three children of the same gender. Assuming P(B) = P(G) = .5, what is the pmf of X = the number of children in the family?

Problem 97E Of all customers purchasing automatic garage-door openers, 75% purchase a chain-driven model. Let X = the number among the next 15 purchasers who select the chain-driven model. a. What is the pmf of X? b. Compute P(X > 10). c. Compute P(6 ? X ? 10). d. Compute ? and ?2. e. If the store currently has in stock 10 chain-driven models and 8 shaft-driven models, what is the probability that the requests of these 15 customers can all be met from existing stock?

Problem 96E The negative binomial rv X was defined as the number of F’s preceding the rth S. Let Y = the number 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.

Problem 98E

Problem 117E A computer disk storage device has ten concentric tracks, numbered 1, 2, . . . , 10 from outermost to innermost, and a single access arm. Let pi = the probability that any particular request for data will take the arm to track . Assume that the tracksi(i = 1, ………,10) accessed in successive seeks are independent. Let X = the number of tracks over which the access arm passes during two successive requests (excluding the track that the arm has just left, so possible X values are x = 0, 1, …..,9). Compute the pmf of X. [Hint: P( the arm is now on track i and X = j) = P(X = j|arm now on i).p.i After the conditional probability is written in terms of p1, . . . , p10, by the law of total probability, the desired probability is obtained by summing over i.]

Problem 118E If X is a hypergeometric rv, show directly from the definition that E(X) = nM/N ( consider only the case n < M). [Hint: Factor nM/N out of the sum for E(X), and show that the terms inside the sum are of the form h( y; n – 1, M – 1, N -1), where y – x -1.]

Problem 119E Use the fact that to prove Chebyshev’s inequality given in Exercise 44.

Problem 13E A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table. Calculate the probability of each of the following events. a. {at most three lines are in use} b. {fewer than three lines are in use} c. {at least three lines are in use} d. {between two and five lines, inclusive, are in use} e. {between two and four lines, inclusive, are not in use} f. {at least four lines are not in use}

Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives circuit boards in batches of five. Two boards are selected from each batch for inspection. We can represent possible outcomes of the selection process by pairs. For example, the pair (1, 2) represents the selection of boards 1 and 2 for inspection. a. List the ten different possible outcomes. b. Suppose that boards 1 and 2 are the only defective boards in a batch. Two boards are to be chosen at random. Define X to be the number of defective boards observed among those inspected. Find the probability distribution of X. c. Let F(x) denote the cdf of X. First determine \(F(0) = P(X \leq 0), F(1)\), and F(2); then obtain F(x) for all other x.

Problem 14E A contractor is required by a county planning department to submit one, two, three, four, or five forms (depending on the nature of the project) in applying for a building permit. Let Y =the number of forms required of the next applicant. The probability that y forms are required is known to be proportional to y—that is,p(y) = ky for y=1,….,5. a. What is the value of k? [Hint: .] b. What is the probability that at most three forms are required? c. What is the probability that between two and four forms (inclusive) are required? d. Could fp(y) = y2 /50 for y = 1…….,5 be the pmf of Y?

A small market orders copies of a certain magazine for its magazine rack each week. Let X = demand for the magazine, with pmf Suppose the store owner actually pays $2.00 for each copy of the magazine and the price to customers is $4.00. If magazines left at the end of the week have no salvage value, is it better to order three or four copies of the magazine? [Hint: For both three and four copies ordered, express net revenue as a function of demand X, and then compute the expected revenue.]

Problem 36E Let X be the damage incurred (in $) in a certain type of accident during a given year. Possible X values are 0, 1000, 5000, and 10000, with probabilities .8, .1, .08, and .02, respectively. A particular company offers a $500 deductible policy. If the company wishes its expected profit to be $100, what premium amount should it charge?

The n candidates for a job have been ranked 1,2,3, . . ., n. Let X = the rank of a randomly selected candidate, so that X has pmf \(p(x)= \begin{cases}1 / n & x=1,2,3, \ldots, n \\ 0 & \text { otherwise }\end{cases}\) (this is called the discrete uniform distribution). Compute E(X) and V(X) using the shortcut formula. [Hint: The sum of the first n positive integers is n(n+1) / 2, whereas the sum of their squares is n(n+1)(2 n+1) / 6.]

Problem 56E 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 probability that exactly 1 received a special accommodation? b. What is the probability that at least 1 received a special accommodation? c. What is the probability that at least 2 received a special accommodation? d. What is the probability that the number among the 25 who received a special accommodation is within 2 standard deviations of the number you would expect to be accommodated? e. Suppose that a student who does not receive a special accommodation is allowed 3 hours for the exam, whereas an accommodated student is allowed 4.5 hours. What would you expect the average time allowed the 25 selected students to be?

Problem 57E A certain type of flashlight requires two type-D batteries, and the flashlight will work only if both its batteries have acceptable voltages. Suppose that 90% of all batteries from a certain supplier have acceptable voltages. Among ten randomly selected flashlights, what is the probability that at least nine will work? What assumptions did you make in the course of answering the question posed?

Problem 58E 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 sample is at most 2. a. What is the probability that the batch will be accepted when the actual proportion of defectives is .01? .05? .10? .20? .25? b. Let p denote the actual proportion of defectives in the batch. A graph of P(batch is accepted) as a function of p, with p on the horizontal axis and P(batch is accepted) on the vertical axis, is called the operating characteristic curve for the acceptance sampling plan. Use the results of part (a) to sketch this curve for 0 ? p ? 1. c. Repeat parts (a) and (b) with “1” replacing “2” in the acceptance sampling plan. d. Repeat parts (a) and (b) with “15” replacing “10” in the acceptance sampling plan. e. Which of the three sampling plans, that of part (a), (c), or (d), appears most satisfactory, and why?

Problem 77E Three brothers and their wives decide to have children until each family has two female children. What is the pmf of X = the number 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?

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 length Y is the number of consecutive time intervals in which the water supply remains below a critical value \(y_0\) (a deficit), preceded by and followed by periods in which the supply exceeds this critical value (a surplus). The cited paper proposes a geometric distribution with p = .409 for this random variable. a. What is the probability that a drought lasts exactly 3 intervals? At most 3 intervals? b. What is the probability that the length of a drought exceeds its mean value by at least one standard deviation?

The article “Expectation Analysis of the Probability of Failure for Water Supply Pipes” (J. of Pipeline Systems Engr. and Practice, May 2012: 36–46) proposed using the Poisson distribution to model the number of failures in pipelines of various types. Suppose that for a cast-iron pipe of a particular length, the expected number of failures is 1 (very close to one of the cases considered in the article). Then X, the number of failures, has a Poisson distribution with \(\mu = 1\). a. Obtain \(P(X \leq 5)\) by using Appendix Table A.2. b. Determine P(X = 2) first from the pmf formula and then from Appendix Table A.2. c. Determine \(P(2 \leq X \leq 4)\). d. What is the probability that X exceeds its mean value by more than one standard deviation?

Problem 99E A k-out-of-n system is one that will function if and only if at least k of the n individual components in the system function. If individual components function independently of one another, each with probability .9, what is the probability that a 3-out-of-5 system functions?

Problem 101E Of the people passing through an airport metal detector, .5% activate it; let X = the number among a randomly selected group of 500 who activate the detector. a. What is the (approximate) pmf of X? b. Compute P(X = 5). c. ComputeP(5 ? X)

Problem 100E A manufacturer of integrated circuit chips wishes to control the quality of its product by rejecting any batch in which the proportion of defective chips is too high. To this end, out of each batch (10,000 chips), 25 will be selected and tested. If at least 5 of these 25 are defective, the entire batch will be rejected. a. What is the probability that a batch will be rejected if 5% of the chips in the batch are in fact defective? b. Answer the question posed in (a) if the percentage of defective chips in the batch is 10%. c. Answer the question posed in (a) if the percentage of defective chips in the batch is 20%. d. What happens to the probabilities in (a)–(c) if the critical rejection number is increased from 5 to 6?

Problem 120E 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 interval [t1, t2] has a Poisson distribution with parameter The occurrence of events over time in this situation is called a nonhomogeneous Poisson process. The article “Inference Based on Retrospective Ascertainment,” J. Amer. Stat. Assoc., 1989: 360–372, considers the intensity function ?(t) = ea+bt as appropriate for events involving transmission of HIV (the AIDS virus) via blood transfusions. Suppose that a = 2 and b = .6(close to values suggested in the paper), with time in years. a. What is the expected number of events in the interval [0, 4]? In [2, 6]? b. What is the probability that at most 15 events occur in the interval [0, .9907]?

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 E(x|Ai) denotes the expected value of X given that the event Ai occurs. Then it can be shown that E(X) = ?E(X|Ai ).P(Ai) the weighted average of the individual “conditional expectations” where the weights are the probabilities of the partitioning events. a. The expected duration of a voice call to a particular telephone number is 3 minutes, whereas the expected duration of a data call to that same number is 1 minute. If 75% of all calls are voice calls, what is the expected duration of the next call? b. A deli sells three different types of chocolate chip cookies. The number of chocolate chips in a type i cookie has a Poisson distribution with parameter ?i = i + 1 (i = 1, 2, 3). If 20% of all customers purchasing a chocolate chip cookie select the first type, 50% choose the second type, and the remaining 30%opt for the third type, what is the expected number of chips in a cookie purchased by the next customer?

Problem 122E 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 transmitted a maximum of 10 times. Assuming that the results of successive transmissions are independent of one another and that the probability of any particular transmission being successful is p, determine the probability mass function of the rv X = the number of times a packet is transmitted. Then obtain an expression for the expected number of times a packet is transmitted.

Consider a deck consisting of seven cards, marked 1, 2,, 7. Three of these cards are selected at random. Define an rv W by W = the sum 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 outcomes. Then there are 35 outcomes, and they can be listed. (This type of rv actually arises in connection with a statistical procedure called Wilcoxons rank-sum test, in which there is an x sample and a y sample and W is the sumof the ranks of the xs in the combined sample; see Section15.2.)

After shuffling a deck of 52 cards, a dealer deals out 5. Let X 5 the number of suits represented in the five-card hand. a. Show that the pmf of X is x 1 2 3 4 p(x) .002 .146 .588 .264 [Hint: p(1) = 4P(all are spades), p(2) = 6P(only spades and hearts with at least one of each suit), and p(4) = 4P(2 spades one of each other suit).] b. Compute m, s2 , and s.

The negative binomial rv X was defined as the number of Fs preceding the rth S. Let Y = the number 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

Of all customers purchasing automatic garage-door openers, 75% purchase a chain-driven model. Let X = the number among the next 15 purchasers who select the chain-driven model. a. What is the pmf of X? b. Compute P(X . 10). c. Compute P(6 # X # 10). d. Compute m and s2 . e. If the store currently has in stock 10 chain-driven models and 8 shaft-driven models, what is the probability that the requests of these 15 customers can all be met from existing stock?

In some applications the distribution of a discrete rv X resembles the Poisson distribution except that zero is not a possible value of X. For example, let X 5 the number of tattoos that an individual wants removed when she or he arrives at a tattoo-removal facility. Suppose the pmf of X is p(x) = k e2u ux x x = 1, 2, 3, a. Determine the value of k. Hint: The sum of all probabilities in the Poisson pmf is 1, and this pmf must also sum to 1. b. If the mean value of X is 2.313035, what is the probability that an individual wants at most 5 tattoos removed? c. Determine the standard deviation of X when the mean value is as given in (b). [Note: The article An Exploratory Investigation of Identity Negotiation and Tattoo Removal (Academy of Marketing Science Review, vol. 12, no. 6, 2008) gave a sample of 22 observations on the number of tattoos people wanted removed; estimates of m and s calculated from the data were 2.318182 and 1.249242, respectively.]

A k-out-of-n system is one that will function if and only if at least k of the n individual components in the system function. If individual components function independently of one another, each with probability .9, what is the probability that a 3-out-of-5 system functions?

A manufacturer of integrated circuit chips wishes to control the quality of its product by rejecting any batch in which the proportion of defective chips is too high. To this end, out of each batch (10,000 chips), 25 will be selected and tested. If at least 5 of these 25 are defective, the entire batch will be rejected. a. What is the probability that a batch will be rejected if 5% of the chips in the batch are in fact defective? b. Answer the question posed in (a) if the percentage of defective chips in the batch is 10%. c. Answer the question posed in (a) if the percentage of defective chips in the batch is 20%. d. What happens to the probabilities in (a)(c) if the critical rejection number is increased from 5 to 6?

Of the people passing through an airport metal detector, .5% activate it; let X = the number among a randomly selected group of 500 who activate the detector. a. What is the (approximate) pmf of X? b. Compute P(X = 5). c. Compute \(P(5 \leq X)\).

An educational consulting firm is trying to decide whether high school students who have never before used a hand-held 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 25students is selected and allowed to practice on both calculators. Then each student is asked to work one problem on the reverse Polish calculator and a similar problem on the other. Let p = P(S), where S indicates that a student worked the problem more quickly using reverse Polish logic than without, and let X 5 number of Ss. a. If p = .5, what is P(7 # X # 18)? b. If p = .8, what is P(7 # X # 18)? c. If the claim that p = .5 is to be rejected when either x # 7 or x $ 18, what is the probability of rejecting the claim when it is actually correct? d. If the decision to reject the claim p = .5 is made as in part (c), what is the probability that the claim is not rejected when p = .6? When p = .8? e. What decision rule would you choose for rejecting the claim p = .5 if you wanted the probability in part (c) to be at most .01?

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 samples. A potentially more economical approach, group testing, was introduced during World War II to identify syphilitic men among army inductees. First, take a part of each blood sample, combine these specimens, and carry out a single test. If no one has the disease, the result will be negative, and only the one test is required. If at least one individual is diseased, the test on the combined sample will yield a positive result, in which case the n individual tests are then carried out. If p = .1 and n = 3, what is the expected number of tests using this procedure? What is the expected number when n = 5? [The article Random Multiple-Access Communication and Group Testing (IEEE Trans. on Commun., 1984: 769774) applied these ideas to a communication system in which the dichotomy was active/ idle user rather than diseased/nondiseased.]

Let p1 denote the probability that any particular code symbol is erroneously transmitted through a communication system. Assume that on different symbols, errors occur independently of one another. Suppose also that with probability p2 an erroneous symbol is corrected upon receipt. Let X denote the number of correct symbols in a message block consisting of n symbols (after the correction process has ended). What is the probability distribution of X?

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 startups are independent of one another. Let p denote the probability that any particular start-up is successful. The random variable of interest is X = the number of startups that must be made prior to acceptance. Give the pmf of X for the case c = 2. If p = .9, what is P(X # 8)? [Hint: For x $ 5, express p(x) recursively in terms of the pmf evaluated at the smaller values x 2 3, x 2 4, , 2.] (This problem was suggested by the article Evaluation of a Start-Up Demonstration Test, J. Quality Technology, 1983: 103106.)

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) qualify for membership? b. Again assuming the validity of the premise, what are the expected number of customers who qualify and the standard deviation of the number who qualify in a random sample of 100 current customers? c. Let X denote the number in a random sample of 25 current customers who qualify for membership. Consider rejecting the companys premise in favor of the claim that p . .10 if x $ 7. What is the probability that the companys premise is rejected when it is actually valid? d. Refer to the decision rule introduced in part (c). What is the probability that the companys premise is not rejected even though p = .20 (i.e., 20% qualify)?

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 ear with single spikelets 29% of the time, whereas a seed with paired spikelets will produce an ear with single spikelets 26% of the time. Consider randomly selecting ten seeds. a. What is the probability that exactly five of these seeds carry a single spikelet and produce an ear with a single spikelet? b. What is the probability that exactly five of the ears produced by these seeds have single spikelets? What is the probability that at most five ears have single spikelets?

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 is the pmf of X 5 the number of jurors favoring acquittal among those interviewed? How many of those favoring acquittal do you expect to be interviewed?

A reservation service employs five information operators who receive requests for information independently of one another, each according to a Poisson process with rate a = 2 per minute. a. What is the probability that during a given 1-min period, the first operator receives no requests? b. What is the probability that during a given 1-min period, exactly four of the five operators receive no requests? c. Write an expression for the probability that during a given 1-min period, all of the operators receive exactly the same number of requests.

Grasshoppers are distributed at random in a large field according to a Poisson process with parameter a = 2 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 newsstand has ordered five copies of a certain issue of a photography magazine. Let X = the number of individuals who come in to purchase this magazine. If X has a Poisson distribution with parameter m = 4, what is the expected number of copies that are sold?

Individuals A and B begin to play a sequence of chess games. Let S = {A wins a game}, and suppose that outcomes of successive games are independent with P(S) 5 p and P(F) = 1 2 p (they never draw). They will play until one of them wins ten games. Let X = the number of games played (with possible values 10, 11,, 19). a. For x = 10, 11, , 19, obtain an expression for p(x) = P(X 5 x). b. If a draw is possible, with p = P(S), q = P(F), 1 2 p 2 q = P(draw), what are the possible values of X? What is P(20 # X)? [Hint: P(20 # X) = 1 2 P(X , 20).]

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 probability .10 of giving a false-negative result. Suppose that ten individuals are tested, five of whom have the disease and five of whom do not. Let X = the number of positive readings that result. a. Does X have a binomial distribution? Explain your reasoning. b. What is the probability that exactly three of the ten test results are positive?

The generalized negative binomial pmf is given by nb(x; r, p) = k(r, x) ? pr (1 2 p)x x = 0, 1, 2, Let X, the number of plants of a certain species found in a particular region, have this distribution with p = .3 and r = 2.5. What is P(X = 4)? What is the probability that at least one plant is found?

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 has a Poisson distribution with mean value m2, and that each CPA prepares the same number of forms of this type. Then if a form of this type is randomly selected, the function p(x; m1, m2) = .5 e2m1m1 x x! 1 .5 e2m2m2 x x! x = 0, 1, 2, gives the pmf of X = the number of errors on the selected form. a. Verify that p(x; m1, m2) is in fact a legitimate pmf ($ 0 and sums to 1). b. What is the expected number of errors on the selected form? c. What is the variance of the number of errors on the selected form? d. How does the pmf change if the first CPA prepares 60% of all such forms and the second prepares 40%?

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 \(X \sim \operatorname{Bin}(n, p)\). By considering the ratio b(x+1 ; n, p) / b(x ; n, p), show that b(x ; n, p) increases with x as long as \(x<n p-(1-p)\). Conclude that the mode \(x^*\) is the integer satisfying \((n+1) p- 1 \leq x^* \leq(n+1) p\). b. Show that if X has a Poisson distribution with parameter \(\mu\), the mode is the largest integer less than \(\mu\). If \(\mu\) is an integer, show that both \(\mu-1\) and \(\mu\) are modes.

A computer disk storage device has ten concentric tracks, numbered 1, 2,, 10 from outermost to innermost, and a single access arm. Let pi = the probability that any particular request for data will take the arm to track i(i = 1, , 10). Assume that the tracks accessed in successive seeks are independent. Let X = the number of tracks over which the access arm passes during two successive requests (excluding the track that the arm has just left, so possible X values are x = 0, 1, , 9). Compute the pmf of X. [Hint: P(the arm is now on track i and X = j) = P(X = j|arm nowon i) ? pi . After the conditional pro bability is written in terms of p1,, p10, by the law of total probability, the desired probability is obtained by summing over i.]

If X is a hypergeometric rv, show directly from the definition that E(X) = nM/N (consider only the case n < M). [Hint: Factor nM/N out of the sum for E(X), and show that the terms inside the sum are of the form h(y; n - 1, M - 1, N - 1), where y = x - 1.]

Use the fact that \(\sum_{\text {all } x}(x-\mu)^2 p(x) \geq \sum_{x:|x-\mu| \geq k \sigma}(x-\mu)^2 p(x)\) to prove Chebyshev's inequality given in Exercise 44 .

The simple Poisson process of Section 3.6 is characterized by a constant rate a 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 [t, t + Dt] is a(t) ? Dt + o(Dt). It can then be shown that the number of events occurring during an interval [t1, t2] has a Poisson distribution with parameter m = # t1 t2 a(t) dt The occurrence of events over time in this situation is called a nonhomogeneous Poisson process. The article Inference Based on Retrospective Ascertainment, (J. Amer. Stat. Assoc., 1989: 360372), considers the intensity function a(t) 5 ea1bt as appropriate for events involving transmission of HIV (the AIDS virus) via blood transfusions. Suppose that a = 2 and b = .6 (close to values suggested in the paper), with time in years. a. What is the expected number of events in the interval [0, 4]? In [2, 6]? b. What is the probability that at most 15 events occur in the interval [0, .9907]?

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 E(XuAi ) denote the expected value of X given that the event Ai occurs. Then it can be shown that E(X) = oE(XuAi ) ? P(Ai ), the weighted average of the individual conditional expectations where the weights are the probabilities of the partitioning events. a. The expected duration of a voice call to a particular telephone number is 3 minutes, whereas the expected duration of a data call to that same number is 1 minute. If 75% of all calls are voice calls, what is the expected duration of the next call? b. A deli sells three different types of chocolate chip cookies. The number of chocolate chips in a type i cookie has a Poisson distribution with parameter mi = i + 1 (i = 1, 2, 3). If 20% of all customers purchasing a chocolate chip cookie select the first type, 50% choose the second type, and the remaining 30% opt for the third type, what is the expected number of chips in a cookie purchased by the next customer?

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 transmitted a maximum of 10 times. Assuming that the results of successive transmissions are independent of one another and that the probability of any particular transmission being successful is p, determine the probability mass function of the rv X = the number of times a packet is transmitted. Then obtain an expression for the expected number of times a packet is transmitted