Find E[Y (Y ? 1)] for a geometric random variable Y by

Problem 1E When the health department tested private wells in a county for two impurities commonly found in drinking water, it found that 20% of the wells had neither impurity, 40% had impurity A, and 50% had impurity B. (Obviously, some had both impurities.) If a well is randomly chosen from those in the county, find the probability distribution for Y , the number of impurities found in the well.

Problem 5E A problem in a test given to small children asks them to match each of three pictures of animals to the word identifying that animal. If a child assigns the three words at random to the three pictures, find the probability distribution for Y , the number of correct matches.

Problem 3E A group of four components is known to contain two defectives. An inspector tests the components one at a time until the two defectives are located. Once she locates the two defectives, she stops testing, but the second defective is tested to ensure accuracy. Let Y denote the number of the test on which the second defective is found. Find the probability distribution for Y .

Consider a system of water flowing through valves from A to B. (See the accompanying diagram.) Valves 1, 2, and 3 operate independently, and each correctly opens on signal with probability .8. Find the probability distribution for Y , the number of open paths from A to B after the signal is given. (Note that Y can take on the values 0, 1, and 2.)

Problem 2E You and a friend play a game where you each toss a balanced coin. If the upper faces on the coins are both tails, you win $1; if the faces are both heads, you win $2; if the coins do not match (one shows a head, the other a tail), you lose $1 (win (?$1)). Give the probability distribution for your winnings, Y , on a single play of this game.

Problem 6E Five balls, numbered 1, 2, 3, 4, and 5, are placed in an urn. Two balls are randomly selected from the five, and their numbers noted. Find the probability distribution for the following: a The largest of the two sampled numbers b The sum of the two sampled numbers

Use the results of Exercises 3.216(c) and 3.217 to show that, for a hypergeometric random variable, \(E[Y(Y-1)]=\frac{r(r-1) n(n-1)}{N(N-1)}\). Equation Transcription: Text Transcription: E[Y(Y-1)]=r(r-1)n(n-1) over N(N-1)

Problem 7E Each of three balls are randomly placed into one of three bowls. Find the probability distribution for Y = the number of empty bowls.

Problem 8E A single cell can either die, with probability .1, or split into two cells, with probability .9, producing a new generation of cells. Each cell in the new generation dies or splits into two cells independently with the same probabilities as the initial cell. Find the probability distribution for the number of cells in the next generation.

Problem 9E In order to verify the accuracy of their financial accounts, companies use auditors on a regular basis to verify accounting entries. The company’s employees make erroneous entries 5% of the time. Suppose that an auditor randomly checks three entries. a Find the probability distribution for Y , the number of errors detected by the auditor. b Construct a probability histogram for p(y). c Find the probability that the auditor will detect more than one error.

Problem 10E A rental agency, which leases heavy equipment by the day, has found that one expensive piece of equipment is leased, on the average, only one day in five. If rental on one day is independent of rental on any other day, find the probability distribution of Y , the number of days between a pair of rentals.

Problem 11E Persons entering a blood bank are such that 1 in 3 have type O+ blood and 1 in 15 have type O?blood. Consider three randomly selected donors for the blood bank. Let X denote the number of donors with type O+ blood and Y denote the number with type O? blood. Find the probability distributions for X and Y. Also find the probability distribution for X + Y, the number of donors who have type O blood.

Let Y be a random variable with \(p(y)\) given in the accompanying table. Find \(E(Y)\), \(E(1 / Y)\), \(E\left(Y^{2}-1\right)\), and \(V(Y)\). y 1 2 3 4 \(p(y)\) .4 .3 .2 .1 Equation Transcription: Text Transcription: p(y) E(Y) E(1/Y) E(Y^2-1) V(Y) p(y)

Problem 13E Refer to the coin-tossing game in Exercise 3.2. Calculate the mean and variance of Y, your winnings on a single play of the game. Note that E(Y) > 0. How much should you pay to play this game if your net winnings, the difference between the payoff and cost of playing, are to have mean 0? Reference You and a friend play a game where you each toss a balanced coin. If the upper faces on the coins are both tails, you win $1; if the faces are both heads, you win $2; if the coins do not match (one shows a head, the other a tail), you lose $1 (win (?$1)). Give the probability distribution for your winnings, Y , on a single play of this game.

Problem 16E The secretary in Exercise 2.121 was given n computer passwords and tries the passwords at random. Exactly one password will permit access to a computer file. Find the mean and the variance of Y, the number of trials required to open the file, if unsuccessful passwords are eliminated (as in Exercise 2.121). Reference A new secretary has been given n computer passwords, only one of which will permit access to a computer file. Because the secretary has no idea which password is correct, he chooses one of the passwords at random and tries it. If the password is incorrect, he discards it and randomly selects another password from among those remaining, proceeding in this manner until he finds the correct password. a What is the probability that he obtains the correct password on the first try? b What is the probability that he obtains the correct password on the second try? The third try? c A security system has been set up so that if three incorrect passwords are tried before the correct one, the computer file is locked and access to it denied. If n = 7, what is the probability that the secretary will gain access to the file?

The maximum patent life for a new drug is 17 years. Subtracting the length of time required by the FDA for testing and approval of the drug provides the actual patent life for the drug—that is, the length of time that the company has to recover research and development costs and to make a profit. The distribution of the lengths of actual patent lives for new drugs is given below: Years, y 3 4 5 6 7 8 9 10 11 12 13 \(p(y)\) .03 .05 .07 .10 .14 .20 .18 .12 .07 .03 .01 a Find the mean patent life for a new drug. b Find the standard deviation of Y = the length of life of a randomly selected new drug. c What is the probability that the value of Y falls in the interval \(\mu \pm 2 \sigma\)? Equation Transcription: Text Transcription: p(y) mu+/-2sigma

Problem 15E Who is the king of late night TV? An Internet survey estimates that, when given a choice between David Letterman and Jay Leno, 52% of the population prefers to watch Jay Leno. Three late night TV watchers are randomly selected and asked which of the two talk show hosts they prefer. a Find the probability distribution for Y, the number of viewers in the sample who prefer Leno. b Construct a probability histogram for p(y). c What is the probability that exactly one of the three viewers prefers Leno? d What are the mean and standard deviation for Y ? e What is the probability that the number of viewers favoring Leno falls within 2 standard deviations of the mean?

Problem 17E Refer to Exercise 3.7. Find the mean and standard deviation for Y = the number of empty bowls. What is the probability that the value of Y falls within 2 standard deviations of the mean? Reference Each of three balls are randomly placed into one of three bowls. Find the probability distribution for Y = the number of empty bowls.

Problem 18E Refer to Exercise 3.8. What is the mean number of cells in the second generation? Reference A single cell can either die, with probability .1, or split into two cells, with probability .9, producing a new generation of cells. Each cell in the new generation dies or splits into two cells independently with the same probabilities as the initial cell. Find the probability distribution for the number of cells in the next generation.

Problem 19E An insurance company issues a one-year $1000 policy insuring against an occurrence A that historically happens to 2 out of every 100 owners of the policy. Administrative fees are $15 per policy and are not part of the company’s “profit.” How much should the company charge for the policy if it requires that the expected profit per policy be $50? [Hint: If C is the premium for the policy, the company’s “profit” is C ?15 if A does not occur and C ?15?1000 if A does occur.]

Problem 20E A manufacturing company ships its product in two different sizes of truck trailers. Each shipment is made in a trailer with dimensions 8 feet × 10 feet × 30 feet or 8 feet × 10 feet × 40 feet. If 30% of its shipments are made by using 30-foot trailers and 70% by using 40-foot trailers, find the mean volume shipped per trailer load. (Assume that the trailers are always full.)

Problem 22E A single fair die is tossed once. Let Y be the number facing up. Find the expected value and variance of Y .

The number N of residential homes that a fire company can serve depends on the distance r (in city blocks) that a fire engine can cover in a specified (fixed) period of time. If we assume that N is proportional to the area of a circle R blocks from the firehouse, then \(N=C \pi R^{2}\), where C is a constant, \(\pi=3.1416\)..., and R, a random variable, is the number of blocks that a fire engine can move in the specified time interval. For a particular fire company, \(C=8\), the probability distribution for R is as shown in the accompanying table, and \(p(r)=0\) for \(r \leq 20\) and \(r \geq 27\). r 21 22 23 24 25 26 \(p(r)\) .05 .20 .30 .25 .15 .05 Find the expected value of N, the number of homes that the fire department can serve. Equation Transcription: Text Transcription: N=CpiR^2 pi=3.1416 C=8 p(r)=0 r</=20 r>/=27 p(r)

Problem 23E In a gambling game a person draws a single card from an ordinary 52-card playing deck. A person is paid $15 for drawing a jack or a queen and $5 for drawing a king or an ace. A person who draws any other card pays $4. If a person plays this game, what is the expected gain?

Problem 24E Approximately 10% of the glass bottles coming off a production line have serious flaws in the glass. If two bottles are randomly selected, find the mean and variance of the number of bottles that have serious flaws.

Problem 25E Two construction contracts are to be randomly assigned to one or more of three firms: I, II, and III. Any firm may receive both contracts. If each contract will yield a profit of $90,000 for the firm, find the expected profit for firm I. If firms I and II are actually owned by the same individual, what is the owner’s expected total profit?

Problem 26E A heavy-equipment salesperson can contact either one or two customers per day with probability 1/3 and 2/3, respectively. Each contact will result in either no sale or a $50,000 sale, with the probabilities .9 and .1, respectively. Give the probability distribution for daily sales. Find the mean and standard deviation of the daily sales.3

Problem 27E A potential customer for an $85,000 fire insurance policy possesses a home in an area that, according to experience, may sustain a total loss in a given year with probability of .001 and a 50% loss with probability .01. Ignoring all other partial losses, what premium should the insurance company charge for a yearly policy in order to break even on all $85,000 policies in this area?

Problem 28E Refer to Exercise 3.3. If the cost of testing a component is $2 and the cost of repairing a defective is $4, find the expected total cost for testing and repairing the lot. Reference You and a friend play a game where you each toss a balanced coin. If the upper faces on the coins are both tails, you win $1; if the faces are both heads, you win $2; if the coins do not match (one shows a head, the other a tail), you lose $1 (win (?$1)). Give the probability distribution for your winnings, Y , on a single play of this game.

If Y is a discrete random variable that assigns positive probabilities to only the positive integers, show that \(E(Y)=\sum_{i=1}^{\infty} P(Y \geq k)\). Equation Transcription: Text Transcription: E(Y)=sum i=1 infinity P(Y>/=k)

Suppose that is a discrete random variable with mean and variance \(\sigma^{2}\) and let \(X=Y+1\). a Do you expect the mean of to be larger than, smaller than, or equal to \(\mu=E(Y)\)? Why? b Use Theorems and to express \(E(X)=E(Y+1)\) in terms of \(\mu=E(Y)\). Does this result agree with your answer to part (a)? c Recalling that the variance is a measure of spread or dispersion, do you expect the variance of to be larger than, smaller than, or equal to \(\sigma^{2}=V(Y)\)? Why? 3. Exercises preceded by an asterisk are optional. d Use Definition and the result in part (b) to show that \(V(X)=E\left\{[X-E(X)]^{2}\right\}=E\left[(Y-\mu)^{2}\right]=\sigma^{2}\); that is, \(X=Y+1\) and have equal variances. Equation Transcription: Text Transcription: sigma^2 X=Y+1 mu=E(Y) E(X)=E(Y+1) mu=E(Y) sigma^2=V(Y) V(X)=E{[X-E(X)]^2}=E[(Y-)^2]=sigma^2 X=Y+1

Suppose that is a discrete random variable with mean \(\mu\) and variance \(\sigma^{2}\) and let \(W=2 Y\). a Do you expect the mean of to be larger than, smaller than, or equal to \(\mu=E(Y)\)? Why? b Use Theorem to express \(E(W)=E(2 Y)\) in terms of \(\mu=E(Y)\). Does this result agree with your answer to part (a)? c Recalling that the variance is a measure of spread or dispersion, do you expect the variance of to be larger than, smaller than, or equal to \(\sigma^{2}=V(Y)\)? Why? d Use Definition and the result in part (b) to show that \(V(W)=E\left\{[W-E(W)]^{2}\right\}=E\left[4(Y-\mu)^{2}\right]=4 \sigma^{2}\); that is, \(W=2 Y\) has variance four times that of . Equation Transcription: Text Transcription: mu sigma^2 W=2Y mu=E(Y) E(W)=E(2Y) mu=E(Y) sigma^2=V(Y) V(W)=E{[W-E(W)]^2}=E[4(Y-mu)^2]=4sigma^2 W=2Y

The manager of a stockroom in a factory has constructed the following probability distribution for the daily demand (number of times used) for a particular tool. y 0 1 2 \(p(y)\) .1 .5 .4 It costs the factory $10 each time the tool is used. Find the mean and variance of the daily cost for use of the tool. Equation Transcription: Text Transcription: p(y)

Problem 36E a A meteorologist in Denver recorded Y = the number of days of rain during a 30-day period. Does Y have a binomial distribution? If so, are the values of both n and p given? b A market research firm has hired operators who conduct telephone surveys. A computer is used to randomly dial a telephone number, and the operator asks the answering person whether she has time to answer some questions. Let Y = the number of calls made until the first person replies that she is willing to answer the questions. Is this a binomial experiment? Explain.

Suppose that is a discrete random variable with mean and variance \(\sigma^{2}\) and let \(U=Y / 10\). a Do you expect the mean of to be larger than, smaller than, or equal to \(\mu=E(Y)\)? Why? b Use Theorem to express \(E(U)=E(Y / 10)\) in terms of \(\mu=E(Y)\). Does this result agree with your answer to part (a)? c Recalling that the variance is a measure of spread or dispersion, do you expect the variance of to be larger than, smaller than, or equal to \(\sigma^{2}=V(Y)\)? Why? d Use Definition and the result in part (b) to show that \(V(U)=E\left\{[U-E(U)]^{2}\right\}=E\left[.01(Y-\mu)^{2}\right]=.01 \sigma^{2}\); that is, \(U=Y / 10\) has variance times that of . Equation Transcription: Text Transcription: mu sigma^2 U=Y/10 mu=E(Y) E(U)=E(Y/10) mu=E(Y) sigma^2=V(Y) V(U)=E{[U-E(U)]^2}=E[.01(Y-mu)^2]=.01sigma^2 U=Y/10

Problem 35E Consider the population of voters described in Example 3.6. Suppose that there are N = 5000 voters in the population, 40% of whom favor Jones. Identify the event favors Jones as a success S. It is evident that the probability of S on trial 1 is .40. Consider the event B that S occurs on the second trial. Then B can occur two ways: The first two trials are both successes or the first trial is a failure and the second is a success. Show that P( B) = .4. What is P( B| the first trial is S)? Does this conditional probability differ markedly from P( B)? Reference Suppose that 40% of a large population of registered voters favor candidate Jones. A random sample of n = 10 voters will be selected, and Y, the number favoring Jones, is to be observed. Does this experiment meet the requirements of a binomial experiment? If each of the ten people is selected at random from the population, then we have ten nearly identical trials, with each trial resulting in a person either favoring Jones (S) or not favoring Jones (F). The random variable of interest is then the number of successes in the ten trials. For the first person selected, the probability of favoring Jones (S) is .4. But what can be said about the unconditional probability that the second person will favor Jones? In Exercise 3.35 you will show that unconditionally the probability that the second person favors Jones is also .4. Thus, the probability of a success S stays the same from trial to trial. However, the conditional probability of a success on later trials depends on the number of successes in the previous trials. If the population of voters is large, removal of one person will not substantially change the fraction of voters favoring Jones, and the conditional probability that the second person favors Jones will be very close to .4. In general, if the population is large and the sample size is relatively small, the conditional probability of success on a later trial given the number of successes on the previous trials will stay approximately the same regardless of the outcomes on previous trials. Thus, the trials will be approximately independent and so sampling problems of this type are approximately binomial.

Let be a discrete random variable with mean \(\mu\) and variance \(\sigma^{2}\). If and are constants, use Theorems through to prove that a \(E(a Y+b)=a E(Y)+b=a \mu+b\). b \(V(a Y+b)=a^{2} V(Y)=a^{2} \sigma^{2}\). Equation Transcription: Text Transcription: sigma2 E(aY+b)=aE(Y)+b=a mu+b V(aY+b)=a2V(Y)=a^2 sigma^2

Problem 37E In 2003, the average combined SAT score (math and verbal) for college-bound students in the United States was 1026. Suppose that approximately 45% of all high school graduates took this test and that 100 high school graduates are randomly selected from among all high school grads in the United States. Which of the following random variables has a distribution that can be approximated by a binomial distribution? Whenever possible, give the values for n and p. a The number of students who took the SAT b The scores of the 100 students in the sample c The number of students in the sample who scored above average on the SAT d The amount of time required by each student to complete the SAT e The number of female high school grads in the sample

Problem 38E The manufacturer of a low-calorie dairy drink wishes to compare the taste appeal of a new formula (formula B) with that of the standard formula (formula A). Each of four judges is given three glasses in random order, two containing formula A and the other containing formula B. Each judge is asked to state which glass he or she most enjoyed. Suppose that the two formulas are equally attractive. Let Y be the number of judges stating a preference for the new formula. a Find the probability function for Y . b What is the probability that at least three of the four judges state a preference for the new formula? c Find the expected value of Y . d Find the variance of Y .

Problem 39E A complex electronic system is built with a certain number of backup components in its subsystems. One subsystem has four identical components, each with a probability of .2 of failing in less than 1000 hours. The subsystem will operate if any two of the four components are operating. Assume that the components operate independently. Find the probability that a exactly two of the four components last longer than 1000 hours. b the subsystem operates longer than 1000 hours.

Problem 42E Refer to Exercise 3.41. What is the probability that a student answers at least ten questions correctly if a for each question, the student can correctly eliminate one of the wrong answers and subsequently answers each of the questions with an independent random guess among the remaining answers? b he can correctly eliminate two wrong answers for each question and randomly chooses from among the remaining answers? Reference A multiple-choice examination has 15 questions, each with five possible answers, only one of which is correct. Suppose that one of the students who takes the examination answers each of the questions with an independent random guess. What is the probability that he answers at least ten questions correctly?

Problem 43E Many utility companies promote energy conservation by offering discount rates to consumers who keep their energy usage below certain established subsidy standards. A recent EPA report notes that 70% of the island residents of Puerto Rico have reduced their electricity usage sufficiently to qualify for discounted rates. If five residential subscribers are randomly selected from San Juan, Puerto Rico, find the probability of each of the following events: a All five qualify for the favorable rates. b At least four qualify for the favorable rates.

Problem 41E A multiple-choice examination has 15 questions, each with five possible answers, only one of which is correct. Suppose that one of the students who takes the examination answers each of the questions with an independent random guess. What is the probability that he answers at least ten questions correctly?

Problem 40E The probability that a patient recovers from a stomach disease is .8. Suppose 20 people are known to have contracted this disease. What is the probability that a exactly 14 recover? b at least 10 recover? c at least 14 but not more than 18 recover? d at most 16 recover?

Problem 44E A new surgical procedure is successful with a probability of p. Assume that the operation is performed five times and the results are independent of one another. What is the probability that a all five operations are successful if p = .8? b exactly four are successful if p = .6? c less than two are successful if p = .3?

A fire-detection device utilizes three temperature-sensitive cells acting independently of each other in such a manner that any one or more may activate the alarm. Each cell possesses a probability of \(p=.8\) of activating the alarm when the temperature reaches \(100^{\circ}\) Celsius or more. Let Y equal the number of cells activating the alarm when the temperature reaches \(100^{\circ}\). a Find the probability distribution for Y . b Find the probability that the alarm will function when the temperature reaches \(100^{\circ}\). Equation Transcription: Text Transcription: p=8 100deg 100deg 100deg

Problem 49E A manufacturer of floor wax has developed two new brands, A and B, which she wishes to subject to homeowners’ evaluation to determine which of the two is superior. Both waxes, A and B, are applied to floor surfaces in each of 15 homes. Assume that there is actually no difference in the quality of the brands. What is the probability that ten or more homeowners would state a preference for a brand A? b either brand A or brand B?

Problem 51E In the 18th century, the Chevalier de Mere asked Blaise Pascal to compare the probabilities of two events. Below, you will compute the probability of the two events that, prior to contrary gambling experience, were thought by de Mere to be equally likely. a What is the probability of obtaining at least one 6 in four rolls of a fair die? b If a pair of fair dice is tossed 24 times, what is the probability of at least one double six?

Problem 48E A missile protection system consists of n radar sets operating independently, each with a probability of .9 of detecting a missile entering a zone that is covered by all of the units. a If n = 5 and a missile enters the zone, what is the probability that exactly four sets detect the missile? At least one set? b How large must n be if we require that the probability of detecting a missile that enters the zone be .999?

Problem 50E In Exercise 2.151, you considered a model for the World Series. Two teams A and B playa series of games until one team wins four games. We assume that the games are played independently and that the probability that A wins any game is p. Compute the probability that the series lasts exactly five games. [Hint: Use what you know about the random variable, Y, the number of games that A wins among the first four games.] Reference A Model for the World Series Two teams A and B play a series of games until one team wins four games. We assume that the games are played independently and that the probability that A wins any game is p. What is the probability that the series lasts exactly five games?

Problem 53E Tay-Sachs disease is a genetic disorder that is usually fatal in young children. If both parents are carriers of the disease, the probability that their offspring will develop the disease is approximately .25. Suppose that a husband and wife are both carriers and that they have three children. If the outcomes of the three pregnancies are mutually independent, what are the probabilities of the following events? a All three children develop Tay-Sachs. b Only one child develops Tay-Sachs. c The third child develops Tay-Sachs, given that the first two did not.

Problem 52E The taste test for PTC (phenylthiocarbamide) is a favorite exercise in beginning human genetics classes. It has been established that a single gene determines whether or not an individual is a “taster.” If 70% of Americans are “tasters” and 20 Americans are randomly selected, what is the probability that a at least 17 are “tasters”? b fewer than 15 are “tasters”?

Use Table 1, Appendix 3, to construct a probability histogram for the binomial probability distribution for \(n=20\) and \(p=.5\). Notice that almost all the probability falls in the interval \(5 \leq y \leq 15\). Equation Transcription: Text Transcription: n=20 p=.5 5</=y</=15

Suppose that is a binomial random variable based on trials with success probability and consider \(Y^{\star}=n-Y\). a Argue that for \(y^{\star}=0,1, \ldots, n\) \(P\left(Y^{\star}=y^{\star}\right)=P\left(n-Y=y^{\star}\right)=P\left(Y=n-y^{\star}\right)\). b Use the result from part (a) to show that \(P\left(Y^{\star}=y^{\star}\right)=\left(\begin{array}{c} n \\ n-y^{\star} \end{array}\right) p^{n-y^{\star}} q^{y^{\star}}=\left(\begin{array}{l} n \\ y^{\star} \end{array}\right) q^{y^{\star}} p^{n-y^{\star}} \) c The result in part (b) implies that \(Y^{\star}\) has a binomial distribution based on trials and "success" probability \(p^{\star}=q=1-p\). Why is this result "obvious"? Equation Transcription: Text Transcription: Y=n-Y y^star=0,1,...,n P(Y^star=y^star)=P(n-Y=y^star)=P(Y=n-y^star) P(Y^star=y^star)=(_n-y^star ^n)p^n-y^ starq^y^star=(_y^star ^n)q^y^starp^n-y^star Y^star p^star=q=1-p

Construct probability histograms for the binomial probability distributions for \(n=5\), \(p=.1\), .5, and .9. (Table 1, Appendix 3, will reduce the amount of calculation.) Notice the symmetry for \(p=.5\) and the direction of skewness for \(p=.1\) and . Equation Transcription: Text Transcription: n=5 p=.1 p=.5 p=.1

Problem 56E An oil exploration firm is formed with enough capital to finance ten explorations. The probability of a particular exploration being successful is .1. Assume the explorations are independent. Find the mean and variance of the number of successful explorations.

Problem 59E Ten motors are packaged for sale in a certain warehouse. The motors sell for $100 each, but a double-your-money-back guarantee is in effect for any defectives the purchaser may receive. Find the expected net gain for the seller if the probability of any one motor being defective is .08. (Assume that the quality of any one motor is independent of that of the others.)

A particular concentration of a chemical found in polluted water has been found to be lethal to 20% of the fish that are exposed to the concentration for 24 hours. Twenty fish are placed in a tank containing this concentration of chemical in water. a Find the probability that exactly 14 survive. b Find the probability that at least 10 survive. c Find the probability that at most 16 survive. d Find the mean and variance of the number that survive.

Problem 60E A particular concentration of a chemical found in polluted water has been found to be lethal to 20% of the fish that are exposed to the concentration for 24 hours. Twenty fish are placed in a tank containing this concentration of chemical in water. a Find the probability that exactly 14 survive. b Find the probability that at least 10 survive. c Find the probability that at most 16 survive. d Find the mean and variance of the number that survive.

Problem 61E Of the volunteers donating blood in a clinic, 80% have the Rhesus (Rh) factor present in their blood. a If five volunteers are randomly selected, what is the probability that at least one does not have the Rh factor? b If five volunteers are randomly selected, what is the probability that at most four have the Rh factor? c What is the smallest number of volunteers who must be selected if we want to be at least 90% certain that we obtain at least five donors with the Rh factor?

Suppose that is a binomial random variable with \(n>2\) trials and success probability . Use the technique presented in Theorem and the fact that \(E\{(Y-1)(Y-2)\}=E\left(Y^{3}\right)-3 E\left(Y^{2}\right)+2 E(Y)\) to derive \(E\left(Y^{3}\right)\). Equation Transcription: Text Transcription: n>2 E{(Y-1)(Y-2)}=E(Y^3)-3E(Y^2)+2E(Y) E(Y^3)

Problem 62E Goranson and Hall (1980) explain that the probability of detecting a crack in an airplane wing is the product of p1, the probability of inspecting a plane with a wing crack; p2, the probability of inspecting the detail in which the crack is located; and p3, the probability of detecting the damage. a What assumptions justify the multiplication of these probabilities? b Suppose p1 = .9, p2 = .8, and p3 = .5 for a certain fleet of planes. If three planes are inspected from this fleet, find the probability that a wing crack will be detected on at least one of them.

Suppose that is a random variable with a geometric distribution. Show that a \(\Sigma_{y} p(y)=\Sigma_{y=1}^{\infty} q^{y-1} p=1\). b \(\frac{p(y)}{p(y-1)}=q\), for \(y=2\), 3,.... This ratio is less than 1 , implying that the geometric probabilities are monotonically decreasing as a function of If has a geometric distribution, what value of is the most likely (has the highest probability)? Equation Transcription: Text Transcription: Sigma_y p(y)=Sigma_y=1^infinity q^y-1 p=1 p(y) over p(y-1)=q y=2

Problem 67E Suppose that 30%of the applicants for a certain industrial job possess advanced training in computer programming. Applicants are interviewed sequentially and are selected at random from the pool. Find the probability that the first applicant with advanced training in programming is found on the fifth interview.

Refer to Exercise 3.56. Suppose the firm has a fixed cost of $20,000 in preparing equipment prior to doing its first exploration. If each successful exploration costs $30,000 and each unsuccessful exploration costs $15,000, find the expected total cost to the firm for its ten explorations.

Refer to Exercise 3.64. The maximum likelihood estimator for p is \(Y / n\) (note that Y is the binomial random variable, not a particular value of it). a Derive \(E(Y / n)\). In Chapter 9, we will see that this result implies that \(Y / n\) is an unbiased estimator for p. b Derive \(V(Y / n)\). What happens to \(V(Y / n)\) as n gets large? Equation Transcription: Text Transcription: Y/n E(Y/n) Y/n V(Y/n) V(Y/n)

Problem 68E Refer to Exercise 3.67. What is the expected number of applicants who need to be interviewed in order to find the first one with advanced training? Reference Suppose that 30%of the applicants for a certain industrial job possess advanced training in computer programming. Applicants are interviewed sequentially and are selected at random from the pool. Find the probability that the first applicant with advanced training in programming is found on the fifth interview.

Problem 69E About six months into George W. Bush’s second term as president, a Gallup poll indicated that a near record (low) level of 41%of adults expressed “a great deal” or “quite a lot” of confidence in the U.S. Supreme Court (http://www.gallup.com/poll/content/default.aspx?ci=17011, June 2005). Suppose that you conducted your own telephone survey at that time and randomly called people and asked them to describe their level of confidence in the Supreme Court. Find the probability distribution for Y , the number of calls until the first person is found who does not express “a great deal” or “quite a lot” of confidence in the U.S. Supreme Court.

Problem 70E An oil prospector will drill a succession of holes in a given area to find a productive well. The probability that he is successful on a given trial is .2. a What is the probability that the third hole drilled is the first to yield a productive well? b If the prospector can afford to drill at most ten wells, what is the probability that he will fail to find a productive well?

Consider an extension of the situation discussed in Example If there are trials in a binomial experiment and we observe \(y_{0}\) "successes," show that \(P\left(Y=y_{0}\right)\) is maximized when \(p=y_{0} / n\). Again, we are determining (in general this time) the value of that maximizes the probability of the value of that we actually observed. Equation Transcription: Text Transcription: y_0 P(Y=y_0) p=y_0/n

Problem 73E A certified public accountant (CPA) has found that nine of ten company audits contain substantial errors. If the CPA audits a series of company accounts, what is the probability that the first account containing substantial errors a is the third one to be audited? b will occur on or after the third audited account?

Problem 71E Let Y denote a geometric random variable with probability of success p. a Show that for a positive integer a, P ( Y > a ) = qa . b Show that for positive integers a and b, P ( Y > a + b|Y > a) = qb = P(Y > b). This result implies that, for example, P(Y > 7|Y > 2) = P(Y > 5). Why do you think this property is called the memoryless property of the geometric distribution? c In the development of the distribution of the geometric random variable, we assumed that the experiment consisted of conducting identical and independent trials until the first success was observed. In light of these assumptions, why is the result in part (b) “obvious”?

Problem 74E Refer to Exercise 3.73. What are the mean and standard deviation of the number of accounts that must be examined to find the first one with substantial errors? Reference A certified public accountant (CPA) has found that nine of ten company audits contain substantial errors. If the CPA audits a series of company accounts, what is the probability that the first account containing substantial errors a is the third one to be audited? b will occur on or after the third audited account?

Problem 75E The probability of a customer arrival at a grocery service counter in any one second is equal to .1. Assume that customers arrive in a random stream and hence that an arrival in any one second is independent of all others. Find the probability that the first arrival a will occur during the third one-second interval. b will not occur until at least the third one-second interval.

Problem 72E Given that we have already tossed a balanced coin ten times and obtained zero heads, what is the probability that we must toss it at least two more times to obtain the first head?

Problem 76E If Y has a geometric distribution with success probability .3, what is the largest value, y0, such that P(Y > y0) ? .1?

If Y has a geometric distribution with success probability p, show that \(P(Y=\text { an odd integer })=\frac{p}{1-q^{2}}\). Equation Transcription: Text Transcription: P(Y=an odd integer)=p over 1-q^2

Problem 78E Of a population of consumers, 60% are reputed to prefer a particular brand, A, of toothpaste. If a group of randomly selected consumers is interviewed, what is the probability that exactly five people have to be interviewed to encounter the first consumer who prefers brand A? At least five people?

Problem 79E In responding to a survey question on a sensitive topic (such as “Have you ever tried marijuana?”), many people prefer not to respond in the affirmative. Suppose that 80% of the population have not tried marijuana and all of those individuals will truthfully answer no to your question. The remaining 20% of the population have tried marijuana and 70% of those individuals will lie. Derive the probability distribution of Y , the number of people you would need to question in order to obtain a single affirmative response.

Problem 81E How many times would you expect to toss a balanced coin in order to obtain the first head?

Problem 80E Two people took turns tossing a fair die until one of them tossed a 6. Person A tossed first, B second, A third, and so on. Given that person B threw the first 6, what is the probability that B obtained the first 6 on her second toss (that is, on the fourth toss overall)?

Problem 82E Refer to Exercise 3.70. The prospector drills holes until he finds a productive well. How many holes would the prospector expect to drill? Interpret your answer intuitively. An oil prospector will drill a succession of holes in a given area to find a productive well. The probability that he is successful on a given trial is .2. a What is the probability that the third hole drilled is the first to yield a productive well? b If the prospector can afford to drill at most ten wells, what is the probability that he will fail to find a productive well?

Problem 83E The secretary in Exercises 2.121 and 3.16 was given n computer passwords and tries the passwords at random. Exactly one of the passwords permits access to a computer file. Suppose now that the secretary selects a password, tries it, and—if it does not work—puts it back in with the other passwords before randomly selecting the next password to try (not a very clever secretary!). What is the probability that the correct password is found on the sixth try? Reference A new secretary has been given n computer passwords, only one of which will permit access to a computer file. Because the secretary has no idea which password is correct, he chooses one of the passwords at random and tries it. If the password is incorrect, he discards it and randomly selects another password from among those remaining, proceeding in this manner until he finds the correct password. a What is the probability that he obtains the correct password on the first try? b What is the probability that he obtains the correct password on the second try? The third try? c A security system has been set up so that if three incorrect passwords are tried before the correct one, the computer file is locked and access to it denied. If n = 7, what is the probability that the secretary will gain access to the file? 3.16 The secretary in Exercise 2.121 was given n computer passwords and tries the passwords at random. Exactly one password will permit access to a computer file. Find the mean and the variance of Y, the number of trials required to open the file, if unsuccessful passwords are eliminated (as in Exercise 2.121).

Find \(E[Y(Y-1)]\) for a geometric random variable by finding \(d^{2} / d q^{2}\left(\Sigma_{y=1}^{\infty} q^{y}\right)\). Use this result to find the variance of . Equation Transcription: Text Transcription: E[Y(Y-1)] d^2/dq^2(Sigma_y=1^infinity q^y)

Consider an extension of the situation discussed in Example 3.13. If we observe as the value for a geometric random variable , show that \(P\left(Y=Y_{0}\right)\) is maximized when \(p=1 / y_{0}\). Again, we are determining (in general this time) the value of that maximizes the probability of the value of that we actually observed. Equation Transcription: Text Transcription: P(Y=Y_0) p=1/y_0

Problem 84E Refer to Exercise 3.83. Find the mean and the variance of Y , the number of the trial on which the correct password is first identified. Reference The secretary in Exercises 2.121 and 3.16 was given n computer passwords and tries the passwords at random. Exactly one of the passwords permits access to a computer file. Suppose now that the secretary selects a password, tries it, and—if it does not work—puts it back in with the other passwords before randomly selecting the next password to try (not a very clever secretary!). What is the probability that the correct password is found on the sixth try? Reference A new secretary has been given n computer passwords, only one of which will permit access to a computer file. Because the secretary has no idea which password is correct, he chooses one of the passwords at random and tries it. If the password is incorrect, he discards it and randomly selects another password from among those remaining, proceeding in this manner until he finds the correct password. a What is the probability that he obtains the correct password on the first try? b What is the probability that he obtains the correct password on the second try? The third try? c A security system has been set up so that if three incorrect passwords are tried before the correct one, the computer file is locked and access to it denied. If n = 7, what is the probability that the secretary will gain access to the file? 3.16 The secretary in Exercise 2.121 was given n computer passwords and tries the passwords at random. Exactly one password will permit access to a computer file. Find the mean and the variance of Y, the number of trials required to open the file, if unsuccessful passwords are eliminated (as in Exercise 2.121).

Problem 88E If Y is a geometric random variable, define Y ? = Y ? 1. If Y is interpreted as the number of the trial on which the first success occurs, then Y ? can be interpreted as the number of failures before the first success. If Y ? = Y ? 1, P(Y ? = y) = P(Y ? 1 = y) = P(Y = y + 1) for y = 0, 1, 2, . . . . Show that P ( Y ? = y) = q y p, y = 0, 1, 2, . . . . The probability distribution of Y ? is sometimes used by actuaries as a model for the distribution of the number of insurance claims made in a specific time period.

Problem 90E The employees of a firm that manufactures insulation are being tested for indications of asbestos in their lungs. The firm is requested to send three employees who have positive indications of asbestos on to a medical center for further testing. If 40% of the employees have positive indications of asbestos in their lungs, find the probability that ten employees must be tested in order to find three positives.

Problem 89E Refer to Exercise 3.88. Derive the mean and variance of the random variable Y ? a by using the result in Exercise 3.33 and the relationship Y ? = Y ? 1, where Y is geometric. *b directly, using the probability distribution for Y ? given in Exercise 3.88. Reference If Y is a geometric random variable, define Y ? = Y ? 1. If Y is interpreted as the number of the trial on which the first success occurs, then Y ? can be interpreted as the number of failures before the first success. If Y ? = Y ? 1, P(Y ? = y) = P(Y ? 1 = y) = P(Y = y + 1) for y = 0, 1, 2, . . . . Show that P ( Y ? = y) = q y p, y = 0, 1, 2, . . . . The probability distribution of Y ? is sometimes used by actuaries as a model for the distribution of the number of insurance claims made in a specific time period.

Problem 91E Refer to Exercise 3.90. If each test costs $20, find the expected value and variance of the total cost of conducting the tests necessary to locate the three positives. Reference The employees of a firm that manufactures insulation are being tested for indications of asbestos in their lungs. The firm is requested to send three employees who have positive indications of asbestos on to a medical center for further testing. If 40% of the employees have positive indications of asbestos in their lungs, find the probability that ten employees must be tested in order to find three positives.

Refer to Exercise 3.86. The maximum likelihood estimator for \(p\) is \(1 / Y\) (note that \(Y\) is the geometric random variable, not a particular value of it). Derive \(E(1 / Y)\) [Hint If \(|r|<1\), \(\sum_{i=1}^{\infty} r^{i} / i=-\ln (1-r)\).] Equation Transcription: Text Transcription: p 1/Y Y E(1/Y) |r|<1 sum over t=1 ^infty r^i/i=-ln(1-r)

Problem 92E Ten percent of the engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability that the first non-defective engine will be found on the second trial?

Problem 93E Refer to Exercise 3.92. What is the probability that the third non-defective engine will be found a on the fifth trial? b on or before the fifth trial? Reference Ten percent of the engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability that the first non-defective engine will be found on the second trial?

Problem 94E Refer to Exercise 3.92. Find the mean and variance of the number of the trial on which a the first nondefective engine is found. b the third nondefective engine is found. Reference Ten percent of the engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability that the first non-defective engine will be found on the second trial?

Problem 95E Refer to Exercise 3.92. Given that the first two engines tested were defective, what is the probability that at least two more engines must be tested before the first nondefective is found? Reference Ten percent of the engines manufactured on an assembly line are defective. If engines are randomly selected one at a time and tested, what is the probability that the first non-defective engine will be found on the second trial?

Problem 96E The telephone lines serving an airline reservation office are all busy about 60% of the time. a If you are calling this office, what is the probability that you will complete your call on the first try? The second try? The third try? b If you and a friend must both complete calls to this office, what is the probability that a total of four tries will be necessary for both of you to get through?

Problem 99E In a sequence of independent identical trials with two possible outcomes on each trial, S and F, and with P(S) = p, what is the probability that exactly y trials will occur before the r th success?

Problem 97E A geological study indicates that an exploratory oil well should strike oil with probability .2. a What is the probability that the first strike comes on the third well drilled? b What is the probability that the third strike comes on the seventh well drilled? c What assumptions did you make to obtain the answers to parts (a) and (b)? d Find the mean and variance of the number of wells that must be drilled if the company wants to set up three producing wells.

Consider the negative binomial distribution given in Definition . a Show that if \(y \geq r+1\), \(\frac{p(y)}{p(y-1)}-\left(\frac{y-1}{y-r}\right) q\). This establishes a recursive relationship between successive negative binomial probabilities, because \(p(y)=p(y-1) \times\left(\frac{y-1}{y-r}\right) q\). b Show that \(\frac{p(y)}{p(y-1)}=\left(\frac{y-1}{y-r}\right) q>1\) if \(y<\frac{r-q}{1-q}\). Similarly, \(\frac{p(y)}{p(y-1)}<1\) if \(y>\frac{r-q}{1-q}\). c Apply the result in part (b) for the case \(r=7\), \(p=.5\) to determine the values of for which \(p(y)>p(y-1)\). Equation Transcription: Text Transcription: y>=r+1 p(y) over p(y-1)-(y-1 over y-r)q p(y)=p(y-1)x(y-1 over y-r)q p(y) over p(y-1)=(y-1 over y-r)q>1 y<r-q over 1-q p(y) over p(y-1)<1 y>r-q over 1-q r=7 p=.5 p(y)>p(y-1)

Problem 101E a We observe a sequence of independent identical trials with two possible outcomes on each trial, S and F, and with P(S) = p. The number of the trial on which we observe the fifth success, Y, has a negative binomial distribution with parameters r = 5 and p. Suppose that we observe the fifth success on the eleventh trial. Find the value of p that maximizes P(Y = 11). b Generalize the result from part (a) to find the value of p that maximizes P(Y = y0) when Y has a negative binomial distribution with parameters r (known) and p.

If is a negative binomial random variable, define \(Y^{*}=Y-r\). If is interpreted as the number of the trial on which the th success occurs, then \(Y^{*}\) can be interpreted as the number of failures before the th success. a If \(Y^{*}=Y-r\), \(P\left(Y^{*}=y\right)=P(Y-r=y)=P(Y=y+r)\) for \(y=0\), 1, 2,..., show that \(P\left(Y^{*}=y\right)=\left(\begin{array}{c} y+r-1 \\ r-1 \end{array}\right) p^{r} q^{y} \), \(y=0\), 1, 2….. b Derive the mean and variance of the random variable \(Y^{*}\) by using the relationship \(Y^{*}=Y-r\), where is negative binomial and the result in Exercise Equation Transcription: Text Transcription: Y*=Y-r Y* Y*=Y-r P(Y*=y)=P(Y-r=y)=P(Y=y+r) y=0 P(Y*=y)=( _r-1^y+r-1)p^r q^y y=0 Y* Y*=Y-r

Problem 102E An urn contains ten marbles, of which five are green, two are blue, and three are red. Three marbles are to be drawn from the urn, one at a time without replacement. What is the probability that all three marbles drawn will be green?

Problem 103E A warehouse contains ten printing machines, four of which are defective. A company selects five of the machines at random, thinking all are in working condition. What is the probability that all five of the machines are non-defective?

Problem 104E Twenty identical looking packets of white power are such that 15 contain cocaine and 5 do not. Four packets were randomly selected, and the contents were tested and found to contain cocaine. Two additional packets were selected from the remainder and sold by undercover police officers to a single buyer. What is the probability that the 6 packets randomly selected are such that the first 4 all contain cocaine and the 2 sold to the buyer do not?

Problem 106E Refer to Exercise 3.103. The company repairs the defective ones at a cost of $50 each. Find the mean and variance of the total repair cost. Reference A warehouse contains ten printing machines, four of which are defective. A company selects five of the machines at random, thinking all are in working condition. What is the probability that all five of the machines are non-defective?

Problem 107E A group of six software packages available to solve a linear programming problem has been ranked from 1 to 6 (best to worst). An engineering firm, unaware of the rankings, randomly selected and then purchased two of the packages. Let Y denote the number of packages purchased by the firm that are ranked 3, 4, 5, or 6. Give the probability distribution for Y.

Problem 105E In southern California, a growing number of individuals pursuing teaching credentials are choosing paid internships over traditional student teaching programs. A group of eight candidates for three local teaching positions consisted of five who had enrolled in paid internships and three who enrolled in traditional student teaching programs. All eight candidates appear to be equally qualified, so three are randomly selected to fill the open positions. Let Y be the number of internship trained candidates who are hired. a Does Y have a binomial or hypergeometric distribution? Why? b Find the probability that two or more internship trained candidates are hired. c What are the mean and standard deviation of Y ?

Problem 108E A shipment of 20 cameras includes 3 that are defective. What is the minimum number of cameras that must be selected if we require that P(at least 1 defective) ? .8?

Problem 109E Seed are often treated with fungicides to protect them in poor draining, wet environments. A small-scale trial, involving five treated and five untreated seeds, was conducted prior to a large-scale experiment to explore how much fungicide to apply. The seeds were planted in wet soil, and the number of emerging plants were counted. If the solution was not effective and four plants actually sprouted, what is the probability that a all four plants emerged from treated seeds? b three or fewer emerged from treated seeds? c at least one emerged from untreated seeds?

Problem 113E A jury of 6 persons was selected from a group of 20 potential jurors, of whom 8 were African American and 12 were white. The jury was supposedly randomly selected, but it contained only 1 African American member. Do you have any reason to doubt the randomness of the selection?

Problem 110E A corporation is sampling without replacement for n = 3 firms to determine the one from which to purchase certain supplies. The sample is to be selected from a pool of six firms, of which four are local and two are not local. Let Y denote the number of nonlocal firms among the three selected. a P ( Y = 1). b P ( Y ? 1). c P ( Y ? 1).

Problem 112E Used photocopy machines are returned to the supplier, cleaned, and then sent back out on lease agreements. Major repairs are not made, however, and as a result, some customers receive malfunctioning machines. Among eight used photocopiers available today, three are malfunctioning. A customer wants to lease four machines immediately. To meet the customer’s deadline, four of the eight machines are randomly selected and, without further checking, shipped to the customer. What is the probability that the customer receives a no malfunctioning machines? b at least one malfunctioning machine?

Problem 114E Refer to Exercise 3.113. If the selection process were really random, what would be the mean and variance of the number of African American members selected for the jury? Reference A jury of 6 persons was selected from a group of 20 potential jurors, of whom 8 were African American and 12 were white. The jury was supposedly randomly selected, but it contained only 1 African American member. Do you have any reason to doubt the randomness of the selection?

Specifications call for a thermistor to test out at between 9000 and 10,000 ohms at \(25^{\circ}\) Celcius. Ten thermistors are available, and three of these are to be selected for use. Let Y denote the number among the three that do not conform to specifications. Find the probability distributions for Y (in tabular form) under the following conditions: a Two thermistors do not conform to specifications among the ten that are available. b Four thermistors do not conform to specifications among the ten that are available. Equation Transcription: Text Transcription: 25deg

Problem 115E Suppose that a radio contains six transistors, two of which are defective. Three transistors are selected at random, removed from the radio, and inspected. Let Y equal the number of defectives observed, where Y = 0, 1, or 2. Find the probability distribution for Y. Express your results graphically as a probability histogram.

Problem 116E Simulate the experiment described in Exercise 3.115 by marking six marbles or coins so that two represent defectives and four represent non-defectives. Place the marbles in a hat, mix, draw three, and record Y , the number of defectives observed. Replace the marbles and repeat the process until n = 100 observations of Y have been recorded. Construct a relative frequency histogram for this sample and compare it with the population probability distribution (Exercise 3.115). Reference Suppose that a radio contains six transistors, two of which are defective. Three transistors are selected at random, removed from the radio, and inspected. Let Y equal the number of defectives observed, where Y = 0, 1, or 2. Find the probability distribution for Y. Express your results graphically as a probability histogram.

Problem 118E Five cards are dealt at random and without replacement from a standard deck of 52 cards. What is the probability that the hand contains all 4 aces if it is known that it contains at least 3 aces?

Problem 121E Let Y denote a random variable that has a Poisson distribution with mean ? = 2. Find a P ( Y = 4). b P ( Y ? 4). c P ( Y < 4). d P ( Y ? 4|Y ? 2).

Problem 120E The sizes of animal populations are often estimated by using a capture–tag–recapture method. In this method k animals are captured, tagged, and then released into the population. Some time later n animals are captured, and Y , the number of tagged animals among the n, is noted. The probabilities associated with Y are a function of N, the number of animals in the population, so the observed value of Y contains information on this unknown N. Suppose that k = 4 animals are tagged and then released. A sample of n = 3 animals is then selected at random from the same population. Find P(Y = 1) as a function of N. What value of N will maximize P(Y = 1)?

Problem 117E In an assembly-line production of industrial robots, gearbox assemblies can be installed in one minute each if holes have been properly drilled in the boxes and in ten minutes if the holes must be redrilled. Twenty gearboxes are in stock, 2 with improperly drilled holes. Five gearboxes must be selected from the 20 that are available for installation in the next five robots. a Find the probability that all 5 gearboxes will fit properly. b Find the mean, variance, and standard deviation of the time it takes to install these 5 gearboxes.

Problem 119E Cards are dealt at random and without replacement from a standard 52 card deck. What is the probability that the second king is dealt on the fifth card?

Problem 122E Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that a no more than three customers arrive? b at least two customers arrive? c exactly five customers arrive?

Problem 123E The random variable Y has a Poisson distribution and is such that p(0) = p(1). What is p(2)?

Problem 126E Refer to Exercise 3.122. Assume that arrivals occur according to a Poisson process with an average of seven per hour. What is the probability that exactly two customers arrive in the two-hour period of time between a 2:00 P.M. and 4:00 P.M. (one continuous two-hour period)? b 1:00 P.M. and 2:00 P.M. or between 3:00 P.M. and 4:00 P.M. (two separate one-hour periods that total two hours)? Reference Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that a no more than three customers arrive? b at least two customers arrive? c exactly five customers arrive?

Problem 127E The number of typing errors made by a typist has a Poisson distribution with an average of four errors per page. If more than four errors appear on a given page, the typist must retype the whole page. What is the probability that a randomly selected page does not need to be retyped?

Problem 128E Cars arrive at a toll both according to a Poisson process with mean 80 cars per hour. If the attendant makes a one-minute phone call, what is the probability that at least 1 car arrives during the call?

Problem 129E Refer to Exercise 3.128. How long can the attendant’s phone call last if the probability is at least .4 that no cars arrive during the call? Reference Cars arrive at a toll both according to a Poisson process with mean 80 cars per hour. If the attendant makes a one-minute phone call, what is the probability that at least 1 car arrives during the call?

Problem 125E Refer to Exercise 3.122. If it takes approximately ten minutes to serve each customer, find the mean and variance of the total service time for customers arriving during a 1-hour period. (Assume that a sufficient number of servers are available so that no customer must wait for service.) Is it likely that the total service time will exceed 2.5 hours? Reference Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that a no more than three customers arrive? b at least two customers arrive? c exactly five customers arrive?

Problem 130E A parking lot has two entrances. Cars arrive at entrance I according to a Poisson distribution at an average of three per hour and at entrance II according to a Poisson distribution at an average of four per hour. What is the probability that a total of three cars will arrive at the parking lot in a given hour? (Assume that the numbers of cars arriving at the two entrances are independent.)

Problem 132E The mean number of automobiles entering a mountain tunnel per two-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem?

Problem 133E Assume that the tunnel in Exercise 3.132 is observed during ten two-minute intervals, thus giving ten independent observations Y1, Y2, . . . , Y10, on the Poisson random variable. Find the probability that Y > 3 during at least one of the ten two-minute intervals. Reference The mean number of automobiles entering a mountain tunnel per two-minute period is one. An excessive number of cars entering the tunnel during a brief period of time produces a hazardous situation. Find the probability that the number of autos entering the tunnel during a two-minute period exceeds three. Does the Poisson model seem reasonable for this problem?

Problem 131E The number of knots in a particular type of wood has a Poisson distribution with an average of 1.5 knots in 10 cubic feet of the wood. Find the probability that a 10-cubic-foot block of the wood has at most 1 knot.

Problem 136E Increased research and discussion have focused on the number of illnesses involving the organism Escherichia coli (10257:H7), which causes a breakdown of red blood cells and intestinal hemorrhages in its victims (http://www.hsus.org/ace/11831, March 24, 2004). Sporadic outbreaks of E.coli have occurred in Colorado at a rate of approximately 2.4 per 100,000 for a period of two years. a If this rate has not changed and if 100,000 cases from Colorado are reviewed for this year, what is the probability that at least 5 cases of E.coli will be observed? b If 100,000 cases from Colorado are reviewed for this year and the number of E.coli cases exceeded 5, would you suspect that the state’s mean E.coli rate has changed? Explain.

Problem 137E The probability that a mouse inoculated with a serum will contract a certain disease is .2. Using the Poisson approximation, find the probability that at most 3 of 30 inoculated mice will contract the disease.

Problem 140E A store owner has overstocked a certain item and decides to use the following promotion to decrease the supply. The item has a marked price of $100. For each customer purchasing the item during a particular day, the owner will reduce the price by a factor of one-half. Thus, the first customer will pay $50 for the item, the second will pay $25, and so on. Suppose that the number of customers who purchase the item during the day has a Poisson distribution with mean 2. Find the expected cost of the item at the end of the day. [Hint: The cost at the end of the day is 100(1/2)Y, where Y is the number of customers who have purchased the item.]

Problem 141E A food manufacturer uses an extruder (a machine that produces bite-size cookies and snack food) that yields revenue for the firm at a rate of $200 per hour when in operation. However, the extruder breaks down an average of two times every day it operates. If Y denotes the number of breakdowns per day, the daily revenue generated by the machine is R = 1600 ? 50Y 2. Find the expected daily revenue for the extruder.

Consider a binomial experiment for \(n=20\), \(p=.05\). Use Table 1, Appendix 3, to calculate the binomial probabilities for \(y=0\), 1, 2, 3, and 4. Calculate the same probabilities by using the Poisson approximation with \(\lambda=n p\). Compare. Equation Transcription: Text Transcription: n=20 p=.05 y=0 lambda=np

Problem 139E In the daily production of a certain kind of rope, the number of defects per foot Y is assumed to have a Poisson distribution with mean ? = 2. The profit per foot when the rope is sold is given by X, where X = 50 ? 2Y ? Y 2. Find the expected profit per foot.

Let \(p(y)\) denote the probability function associated with a Poisson random variable with mean \(\lambda\) a. Show that the ratio of successive probabilities satisfies \(\frac{p(y)}{p(y-1)}=\frac{\lambda}{y}\), for \(y=1\), 2,.... b. For which values of is \(p(y)>p(y-1)\)? c. Notice that the result in part (a) implies that Poisson probabilities increase for awhile as increases and decrease thereafter. Show that \(p(y)\) maximized when = the greatest integer less than or equal to \(\lambda\). Equation Transcription: Text Transcription: p(y) lambda p(y) over p(y-1)=lambda over y y=1 p(y)>p(y-1) p(y) lambda

Refer to Exercise 3.142 (c). If the number of phone calls to the fire department, , in a day has a Poisson distribution with mean 5.3, what is the most likely number of phone calls to the fire department on any day?

Differentiate the moment-generating function in Exercise 3.145 to find \(E(Y)\) and \(E\left(Y^{2}\right)\). Then find \(V(Y)\). Equation Transcription: Text Transcription: E(Y) E(Y^2) V(Y)

Approximately of silicon wafers produced by a manufacturer have fewer than two large flaws. If , the number of flaws per wafer, has a Poisson distribution, what proportion of the wafers have more than five large flaws? [: Use Table 3, Appendix 3.]

If has a binomial distribution with trials and probability of success , show that the moment-generating function for is \(m(t)=\left(p e^{t}+q\right)^{n}\), where \(q=1-p\). Equation Transcription: Text Transcription: m(t)=(pe^t+q)^n q=1-p

If has a geometric distribution with probability of success , show that the moment-generating function for is \(m(t)=\frac{p e^{t}}{1-q e^{t}}\), where \(q=1-p\). Equation Transcription: Text Transcription: m(t)=pe^t over 1-qe^t q=1-p

Problem 138E Let Y have a Poisson distribution with mean ?. Find E[Y (Y ? 1)] and then use this to show that V (Y ) = ?.

Differentiate the moment-generating function in Exercise to find \(E(Y)\) and \(E\left(Y^{2}\right)\). Then find \(V(Y)\). Equation Transcription: Text Transcription: E(Y) E(Y^2) V(Y)

Refer to Exercises 3.142 and 3.143. If the number of phone calls to the fire department, , in a day has a Poisson distribution with mean 6, show that \(p(5)=p(6)\) so that and are the most likely values for . Equation Transcription: Text Transcription: p(5)=p(6)

Refer to Exercise 3.145. Use the uniqueness of moment-generating functions to give the distribution of a random variable with moment-generating function \(m(t)=\left(.6 e^{t}+.4\right)^{3}\). Equation Transcription: Text Transcription: m(t)=(.6e^t+.4)^3

Refer to Exercise 3.147. Use the uniqueness of moment-generating functions to give the distribution of a random variable with moment-generating function \(m(t)=\frac{.3 e^{t}}{1-.7 e^{t}}\). Equation Transcription: Text Transcription: m(t)=.3e^t over 1-.7e^t

Find the distributions of the random variables that have each of the following moment generating functions: a. \(m(t)=\left[(1 / 3) e^{t}+(2+3)\right]^{5}\). b. \(m(t)=\frac{e^{t}}{2-e^{t}}\). c. \(m(t)=e^{2\left(e^{t}-1\right)}\). Equation Transcription: Text Transcription: m(t)=[(1/3)e^t+(2+3)]^5 m(t)=e^t over 2-e^t m(t)=e^2(e^t-1)

Refer to Exercise 3.145. If has moment-generating function \(m(t)=\left(.7 e^{t}+.3\right)^{10}\), what is \(P(Y \leq 5)\)? Equation Transcription: Text Transcription: m(t)=(.7e^t+.3)^10 P(Y</=5)

Refer to Example 3.23. If has moment-generating function \(m(t)=e^{6\left(e^{t-1}\right)}\), what is \(P(|Y-u| \leq 2 \sigma)\)? Equation Transcription: Text Transcription: m(t)=e^6(e^t-1) P(|Y-u|</=2 sigma)

Problem 155E Let m(t) = (1/6)et + (2/6)e2t + (3/6)e3t. Find the following: a E ( Y ) b V ( Y ) c The distribution of Y

Refer to Exercise 3.153. By inspection, give the mean and variance of the random variables associated with the moment-generating functions given in parts (a), (b), and (c).

Problem 156E Suppose that Y is a random variable with moment-generating function m(t). a What is m(0)? b If W = 3Y, show that the moment-generating function of W is m(3t). c If X = Y ? 2, show that the moment-generating function of X is e?2t m(t).

Refer to Exercise 3.156. a If \(W=3 Y\), use the moment-generating function of \(W\) to show that \(E(W)=3 E(Y)\) and \(V(W)=9 V(Y)\). b If \(X=Y-2\), use the moment-generating function of \(X\) to show that \(E(X)=E(Y)-2\) and \(V(X)=V(Y)\).

Problem 158E If Y is a random variable with moment-generating function m(t) and if W is given by W = aY + b, show that the moment-generating function of W is etbm(at).

Problem 159E Use the result in Exercise 3.158 to prove that, if W = aY + b, then E(W ) = aE(Y ) + b and V (W ) = a2 V (Y ). Reference If Y is a random variable with moment-generating function m(t) and if W is given by W = aY + b, show that the moment-generating function of W is etbm(at).

Let \(r(t)=\ln [m(t)] and r^{(k)}(0)\) denote the \(k \text { th }\) derivative of \(r(t)\) evaluated for \(t=0\). Show that \(r^{(1)}(0)=\mu_{1}^{\prime}=\mu\) and \(r^{(2)}(0)=\mu_{2}^{\prime}-\left(\mu_{1}^{\prime}\right)^{2}=\sigma^{2}\) [Hint: \(m(0)=1\).] Equation Transcription: th Text Transcription: r(t)=ln[m(t)] r^(k)(0) kth r(t) t=0 r^(1)(0)=mu'_1=mu r^(2)(0)=mu'_2-(mu'1)^2=sigma^2 m(0)=1

Refer to Exercises and 3.158. If has a geometric distribution with success probability , consider \(Y^{\star}=Y-1\). Show that the moment-generating function of \(Y^{\star}\) is \(m^{\star}(t)=\frac{p}{1-q e^{t}}\), where \(q=1-p\). Equation Transcription: Text Transcription: Y^star=Y-1 Y^star m^star (t)=p over 1-qe^t q=1-p

Problem 165E Let Y denote a Poisson random variable with mean ?. Find the probability-generating function for Y and use it to find E(Y ) and V (Y ).

Suppose that is a binomial random variable based on trials with success probability and let \(Y^{\star}=n-Y\). a. Use the result in Exercise to show that \(E\left(Y^{\star}\right)=n q\) and \(V\left(Y^{\star}\right)=n p q\), where \(q=1-p\). b. Use the result in Exercise to show that the moment-generating function of \(Y^{\star}\) is \(m^{\star}(t)=\left(q e^{t}+p\right)^{n}\), where \(q=1-p\). c. Based on your answer to part (b), what is the distribution of \(Y^{\star}\)? d. If is interpreted as the number of successes in a sample of size , what is the interpretation of \(Y^{\star}\)? e. Based on your answer in part (d), why are the answers to parts (a), (b), and (c) "obvious"? Equation Transcription: Text Transcription: Y^star=n-Y E(Y^star)=nq V(Y^star)=npq q=1-p Y^star m^star(t)=(qe^t+p)^n q=1-p Y^star Y^star

Use the results of Exercise to find the mean and variance of a Poisson random variable with \(m(t)=e^{5\left(e^{t}-1\right)}\). Notice that \(r(t)\) is easier to differentiate than \(m(t)\) in this case. Equation Transcription: Text Transcription: m(t)=e5(et-1) r(t) m(t)

Problem 164E Let Y denote a binomial random variable with n trials and probability of success p. Find the probability-generating function for Y and use it to find E(Y ).

Problem 166E Refer to Exercise 3.165. Use the probability-generating function found there to find E(Y 3). Reference Let Y denote a Poisson random variable with mean ?. Find the probability-generating function for Y and use it to find E(Y ) and V (Y ).

Problem 167E Let Y be a random variable with mean 11 and variance 9. Using Tchebysheff’s theorem, find a a lower bound for P(6 < Y < 16). b the value of C such that P(|Y ? 11| ? C) ? .09.

Problem 168E Would you rather take a multiple-choice test or a full-recall test? If you have absolutely no knowledge of the test material, you will score zero on a full-recall test. However, if you are given 5 choices for each multiple-choice question, you have at least one chance in five of guessing each correct answer! Suppose that a multiple-choice exam contains 100 questions, each with 5 possible answers, and guess the answer to each of the questions. a What is the expected value of the number Y of questions that will be correctly answered? b Find the standard deviation of Y . c Calculate the intervals ? ± 2? and ? ± 3? . d If the results of the exam are curved so that 50 correct answers is a passing score, are you likely to receive a passing score? Explain.

Problem 172E Refer to Exercise 3.115. Using the probability histogram, find the fraction of values in the population that fall within 2 standard deviations of the mean. Compare your result with that of Tchebysheff’s theorem. Reference Suppose that a radio contains six transistors, two of which are defective. Three transistors are selected at random, removed from the radio, and inspected. Let Y equal the number of defectives observed, where Y = 0, 1, or 2. Find the probability distribution for Y. Express your results graphically as a probability histogram.

This exercise demonstrates that, in general, the results provided by Tchebysheff's theorem cannot be improved upon. Let be a random variable such that \(p(-1)=\frac{1}{18}, p(0)=\frac{16}{18}, p(1)=\frac{1}{18}\) Equation transcription: Text transcription: p(-1)=frac{1}{18}, p(0)=frac{16}{18}, p(1)=frac{1}{18}

Problem 171E For a certain type of soil the number of wireworms per cubic foot has a mean of 100. Assuming a Poisson distribution of wireworms, give an interval that will include at least 5/9 of the sample values of wireworm counts obtained from a large number of 1-cubic-foot samples.

Problem 173E A balanced coin is tossed three times. Let Y equal the number of heads observed. a Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3. b Construct a probability distribution similar to the one in Table 3.1. c Find the expected value and standard deviation of Y, using the formulas E(Y ) = np and V (Y ) = npq. d Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. Repeat for 2 standard deviations. How do your results compare with the results of Tchebysheff’s theorem and the empirical rule?

Problem 174E Suppose that a coin was definitely unbalanced and that the probability of a head was equal to p = .1. Follow instructions (a), (b), (c), and (d) as stated in Exercise 3.173. Notice that the probability distribution loses its symmetry and becomes skewed when p is not equal to 1/2. Reference A balanced coin is tossed three times. Let Y equal the number of heads observed. a Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3. b Construct a probability distribution similar to the one in Table 3.1. c Find the expected value and standard deviation of Y, using the formulas E(Y ) = np and V (Y ) = npq. d Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. Repeat for 2 standard deviations. How do your results compare with the results of Tchebysheff’s theorem and the empirical rule?

Problem 176E A national poll of 549 teenagers (aged 13 to 17) by the Gallop poll (http://gallup.com/conten/ default.aspx?ci=17110), April, 2005) indicated that 85% “think that clothes that display gang symbols” should be banned at school. If teenagers were really evenly split in their opinions regarding banning of clothes that display gang symbols, comment on the probability of observing this survey result (that is, observing 85% or more in a sample of 549 who are in favor of banning clothes that display gang symbols). What assumption must be made about the sampling procedure in order to calculate this probability? [Hint: Recall Tchebysheff’s theorem and the empirical rule.]

Problem 175E In May 2005, Tony Blair was elected to an historic third term as the British prime minister. A Gallop U.K. poll (http://gallup.com/poll/content/default.aspx?ci=1710, June 28, 2005) conducted after Blair’s election indicated that only 32% of British adults would like to see their son or daughter grow up to become prime minister. If the same proportion of Americans would prefer that their son or daughter grow up to be president and 120 American adults are interviewed, a what is the expected number of Americans who would prefer their child grow up to be president? b what is the standard deviation of the number Y who would prefer that their child grow up to be president? c is it likely that the number of Americans who prefer that their child grow up to be president exceeds 40?

Problem 177E For a certain section of a pine forest, the number of diseased trees per acre, Y, has a Poisson distribution with mean ? = 10. The diseased trees are sprayed with an insecticide at a cost of $3 per tree, plus a fixed overhead cost for equipment rental of $50. Letting C denote the total spraying cost for a randomly selected acre, find the expected value and standard deviation for C. Within what interval would you expect C to lie with probability at least .75?

Problem 180SE Four possibly winning numbers for a lottery—AB-4536, NH-7812, SQ-7855, and ZY-3221— arrive in the mail. You will win a prize if one of your numbers matches one of the winning numbers contained on a list held by those conducting the lottery. One first prize of $100,000, two second prizes of $50,000 each, and ten third prizes of $1000 each will be awarded. To be eligible to win, you need to mail the coupon back to the company at a cost of 33¢ for postage. No purchase is required. From the structure of the numbers that you received, it is obvious the numbers sent out consist of two letters followed by four digits. Assuming that the numbers you received were generated at random, what are your expected winnings from the lottery? Is it worth 33¢ to enter this lottery?

Problem 179E Refer to Exercise 3.91. In this exercise, we determined that the mean and variance of the costs necessary to find three employees with positive indications of asbestos poisoning were 150 and 4500, respectively. Do you think it is highly unlikely that the cost of completing the tests will exceed $350? Reference Refer to Exercise 3.90. If each test costs $20, find the expected value and variance of the total cost of conducting the tests necessary to locate the three positives.

Problem 178E It is known that 10% of a brand of television tubes will burn out before their guarantee has expired. If 1000 tubes are sold, find the expected value and variance of Y , the number of original tubes that must be replaced. Within what limits would Y be expected to fall?

Problem 184SE A city commissioner claims that 80% of the people living in the city favor garbage collection by contract to a private company over collection by city employees. To test the commissioner’s claim, 25 city residents are randomly selected, yielding 22 who prefer contracting to a private company. a If the commissioner’s claim is correct, what is the probability that the sample would contain at least 22 who prefer contracting to a private company? b If the commissioner’s claim is correct, what is the probability that exactly 22 would prefer contracting to a private company? c Based on observing 22 in a sample of size 25 who prefer contracting to a private company, what do you conclude about the commissioner’s claim that 80% of city residents prefer contracting to a private company?

Problem 181SE Sampling for defectives from large lots of manufactured product yields a number of defectives, Y , that follows a binomial probability distribution. A sampling plan consists of specifying the number of items n to be included in a sample and an acceptance number a. The lot is accepted if Y ? a and rejected if Y > a. Let p denote the proportion of defectives in the lot. For n = 5 and a = 0, calculate the probability of lot acceptance if (a) p = 0, (b) p = .1, (c) p = .3, (d) p = .5, (e) p = 1.0. A graph showing the probability of lot acceptance as a function of lot fraction defective is called the operating characteristic curve for the sample plan. Construct the operating characteristic curve for the plan n = 5, a = 0. Notice that a sampling plan is an example of statistical inference. Accepting or rejecting a lot based on information contained in the sample is equivalent to concluding that the lot is either good or bad. “Good” implies that a low fraction is defective and that the lot is therefore suitable for shipment.

Problem 183SE A quality control engineer wishes to study alternative sampling plans: n = 5, a = 1 and n = 25, a = 5. On a sheet of graph paper, construct the operating characteristic curves for both plans, making use of acceptance probabilities at p = .05, p = .10, p = .20, p = .30, and p = .40 in each case. a If you were a seller producing lots with fraction defective ranging from p = 0 to p = .10, which of the two sampling plans would you prefer? b If you were a buyer wishing to be protected against accepting lots with fraction defective exceeding p = .30, which of the two sampling plans would you prefer?

Problem 185SE Twenty students are asked to select an integer between 1 and 10. Eight choose either 4, 5 or 6. a If the students make their choices independently and each is as likely to pick one integer as any other, what is the probability that 8 or more will select 4,5 or 6? b Having observed eight students who selected 4, 5, or 6, what conclusion do you draw based on your answer to part (a)?

Problem 187SE Consider the following game: A player throws a fair die repeatedly until he rolls a 2, 3, 4, 5, or 6. In other words, the player continues to throw the die as long as he rolls 1s. When he rolls a “non-1,” he stops. a What is the probability that the player tosses the die exactly three times? b What is the expected number of rolls needed to obtain the first non-1? c If he rolls a non-1 on the first throw, the player is paid $1. Otherwise, the payoff is doubled for each 1 that the player rolls before rolling a non-1. Thus, the player is paid $2 if he rolls a 1 followed by a non-1; $4 if he rolls two 1s followed by a non-1; $8 if he rolls three 1s followed by a non-1; etc. In general, if we let Y be the number of throws needed to obtain the first non-1, then the player rolls (Y ? 1) 1s before rolling his first non-1, and he is paid 2Y ?1 dollars. What is the expected amount paid to the player?

Problem 189SE A starter motor used in a space vehicle has a high rate of reliability and was reputed to start on any given occasion with probability .99999. What is the probability of at least one failure in the next 10,000 starts?

If is a binomial random variable based on trials and success probability , show that \(P(Y>1 \mid Y \geq 1)=\frac{1-(1-p)^{n}-n p(1-p)^{n-1}}{1-(1-p)^{n}}\) Equation transcription: Text transcription: P(Y>1 Y \geq 1)=frac{1-(1-p)^{n}-n p(1-p)^{n-1}}{1-(1-p)^{n}}

Refer to Exercise 3.115. Find , the expected value of Y , for the theoretical population by using the probability distribution obtained in Exercise 3.115. Find the sample mean for the = 100 measurements generated in Exercise 3.116. Does \(\bar{y}\) provide a good estimate of \(\mu)? Equation transcription: Text transcription: bar{y} mu

Problem 186SE Refer to Exercises 3.67 and 3.68. Let Y denote the number of the trial on which the first applicant with computer training was found. If each interview costs $30, find the expected value and variance of the total cost incurred interviewing candidates until an applicant with advanced computer training is found. Within what limits would you expect the interview costs to fall? Reference Suppose that 30%of the applicants for a certain industrial job possess advanced training in computer programming. Applicants are interviewed sequentially and are selected at random from the pool. Find the probability that the first applicant with advanced training in programming is found on the fifth interview. Refer to Exercise 3.67. What is the expected number of applicants who need to be interviewed in order to find the first one with advanced training?

Problem 191SE Find the population variance ? 2 for Exercise 3.115 and the sample variance s2 for Exercise 3.116. Compare. Reference Suppose that a radio contains six transistors, two of which are defective. Three transistors are selected at random, removed from the radio, and inspected. Let Y equal the number of defectives observed, where Y = 0, 1, or 2. Find the probability distribution for Y. Express your results graphically as a probability histogram. 3.116 Simulate the experiment described in Exercise 3.115 by marking six marbles or coins so that two represent defectives and four represent non-defectives. Place the marbles in a hat, mix, draw three, and record Y , the number of defectives observed. Replace the marbles and repeat the process until n = 100 observations of Y have been recorded. Construct a relative frequency histogram for this sample and compare it with the population probability distribution (Exercise 3.115).

Problem 192SE Toss a balanced die and let Y be the number of dots observed on the upper face. Find the mean and variance of Y. Construct a probability histogram, and locate the interval ? ± 2?. Verify that Tchebysheff’s theorem holds.

Problem 193SE Two assembly lines I and II have the same rate of defectives in their production of voltage regulators. Five regulators are sampled from each line and tested. Among the total of ten tested regulators, four are defective. Find the probability that exactly two of the defective regulators came from line I.

Problem 194SE One concern of a gambler is that she will go broke before achieving her first win. Suppose that she plays a game in which the probability of winning is .1 (and is unknown to her). It costs her $10 to play and she receives $80 for a win. If she commences with $30, what is the probability that she wins exactly once before she loses her initial capital?

Problem 196SE Refer to Exercise 3.195. The cost of repairing the imperfections in the weave is $10 per imperfection. Find the mean and standard deviation of the repair cost for an 8-square-yard bolt of the textile. Reference The number of imperfections in the weave of a certain textile has a Poisson distribution with a mean of 4 per square yard. Find the probability that a a 1-square-yard sample will contain at least one imperfection. b 3-square-yard sample will contain at least one imperfection.

Problem 195SE The number of imperfections in the weave of a certain textile has a Poisson distribution with a mean of 4 per square yard. Find the probability that a a 1-square-yard sample will contain at least one imperfection. b 3-square-yard sample will contain at least one imperfection.

Problem 197SE The number of bacteria colonies of a certain type in samples of polluted water has a Poisson distribution with a mean of 2 per cubic centimeter (cm3). a If four 1-cm3 samples are independently selected from this water, find the probability that at least one sample will contain one or more bacteria colonies. b Howmany1-cm3 samples should be selected in order to have a probabilityof approximately .95 of seeing at least one bacteria colony?

Problem 198SE One model for plant competition assumes that there is a zone of resource depletion around each plant seedling. Depending on the size of the zones and the density of the plants, the zones of resource depletion may overlap with those of other seedlings in the vicinity. When the seeds are randomly dispersed over a wide area, the number of neighbors that any seedling has within an area of size A usually follows a Poisson distribution with mean equal to A × d, where d is the density of seedlings per unit area. Suppose that the density of seedlings is four per square meter. What is the probability that a specified seeding has a no neighbors within 1 meter? b at most three neighbors within 2 meters?

Refer to Exercise 3.181. Use Table 1 , Appendix 3 , to construct the operating characteristic curves for the following sampling plans: a. \(n=10, a=0\). b. \(n=10, a=1\). c. \(n=10, a=2\). For each sampling plan, calculate (lot acceptance) for \(p=0, .05, .1, .3, .5\), and Our intuition suggests that sampling plan (a) would be much less likely to accept bad lots than plans (b) and (c). A visual comparison of the operating characteristic curves will confirm this intuitive conjecture. Equation transcription: Text transcription: n=10, a=0 n=10, a=1 n=10, a=2 p=0, .05, .1, .3, .5

Using the fact that \(e^{z}=1+Z+\frac{Z^{2}}{2 !}+\frac{Z^{3}}{3 !}+\frac{z^{4}}{4 !}+\ldots\) expand the moment-generating function for the binomial distribution \(m(t)=\left(1+p e^{t}\right)^{n}\) into a power series in . (Acquire only the low-order terms in .) Identify \(\mu_{i}^{\prime}\) as the coefficient of \(t^{i} / i !\) appearing in the series. Specifically, find \(\mu_{i}^{\prime}\) and \(\mu_{2}^{\prime}\) and compare them with the results of Exercise . Equation transcription: Text transcription: e^{z}=1+Z+frac{Z^{2}}{2 !}+frac{Z^{3}}{3 !}+frac{z^{4}}{4 !}+ldots m(t)=(1+p e^{t})^{n} t^{i} / i ! \mu{2}^{\prime}

Problem 199SE Insulin-dependent diabetes (IDD) is a common chronic disorder in children. The disease occurs most frequently in children of northern European descent, but the incidence ranges from a low of 1–2 cases per 100,000 per year to a high of more than 40 cases per 100,000 in parts of Finland.4 Let us assume that a region in Europe has an incidence of 30 cases per 100,000 per year and that we randomly select 1000 children from this region. a Can the distribution of the number of cases of IDD among those in the sample be approximated by a Poisson distribution? If so, what is the mean of the approximating Poisson distribution? b What is the probability that we will observe at least two cases of IDD among the 1000 children in the sample?

Problem 201SE Refer to Exercises 3.103 and 3.106. In what interval would you expect the repair costs on these five machines to lie? (Use Tchebysheff’s theorem.) Reference 3.103 A warehouse contains ten printing machines, four of which are defective. A company selects five of the machines at random, thinking all are in working condition. What is the probability that all five of the machines are nondefective? 3.106 Refer to Exercise 3.103. The company repairs the defective ones at a cost of $50 each. Find the mean and variance of the total repair cost.

Problem 202SE The number of cars driving past a parking area in a one-minute time interval has a Poisson distribution with mean ?. The probability that any individual driver actually wants to park his or her car is p. Assume that individuals decide whether to park independently of one another. a If one parking place is available and it will take you one minute to reach the parking area, what is the probability that a space will still be available when you reach the lot? (Assume that no one leaves the lot during the one-minute interval.) b Let W denote the number of drivers who wish to park during a one-minute interval. Derive the probability distribution of W .

A type of bacteria cell divides at a constant rate \(\lambda\) over time. (That is, the probability that a cell divides in a small interval of time is approximately ) Given that a population starts out at time zero with cells of this bacteria and that cell divisions are independent of one another, the size of the population at time , has the probability distribution \(P[Y(t)=n]=(n-1 k-1) e^{\lambda k t}\left(1-e^{-\lambda s}\right)^{n-k}, n=k, k+1, \ldots\) a. Find the expected value and variance of in terms of \(\lambda\) and . b. If, for a type of bacteria cell, \(\lambda=1\) per second and the population starts out with two cells at time zero, find the expected value and variance of the population after five seconds. Equation transcription: Text transcription: P[Y(t)=n]=(n-1 k-1) e^{\lambda k t}\left(1-e^{-\lambda s}\right)^{n-k}, n=k, k+1, \ldots lambda lambda=1

Problem 205SE An experiment consists of tossing a fair die until a 6 occurs four times. What is the probability that the process ends after exactly ten tosses with a 6 occurring on the ninth and tenth tosses?

Problem 208SE A recent survey suggests that Americans anticipate a reduction in living standards and that a steadily increasing level of consumption no longer may be as important as it was in the past. Suppose that a poll of 2000 people indicated 1373 in favor of forcing a reduction in the size of American automobiles by legislative means. Would you expect to observe as many as 1373 in favor of this proposition if, in fact, the general public was split 50–50 on the issue? Why?

Problem 204SE The probability that any single driver will turn left at an intersection is .2. The left turn lane at this intersection has room for three vehicles. If the left turn lane is empty when the light turns red and five vehicles arrive at this intersection while the light is red, find the probability that the left turn lane will hold the vehicles of all of the drivers who want to turn left.

Problem 206E Accident records collected by an automobile insurance company give the following information. The probability that an insured driver has an automobile accident is .15. If an accident has occurred, the damage to the vehicle amounts to 20% of its market value with a probability of .80, to 60% of its market value with a probability of .12, and to a total loss with a probability of .08. What premium should the company charge on a $12,000 car so that the expected gain by the company is zero?

Problem 209SE A supplier of heavy construction equipment has found that new customers are normally obtained through customer requests for a sales call and that the probability of a sale of a particular piece of equipment is .3. If the supplier has three pieces of the equipment available for sale, what is the probability that it will take fewer than five customer contacts to clear the inventory?

Problem 207SE The number of people entering the intensive care unit at a hospital on any single day possesses a Poisson distribution with a mean equal to five persons per day. a What is the probability that the number of people entering the intensive care unit on a particular day is equal to 2? Is less than or equal to 2? b Is it likely that Y will exceed 10? Explain.

Problem 211SE A merchant stocks a certain perishable item. She knows that on any given day she will have a demand for either two, three, or four of these items with probabilities .1, .4, and .5, respectively. She buys the items for $1.00 each and sells them for $1.20 each. If any are left at the end of the day, they represent a total loss. How many items should the merchant stock in order to maximize her expected daily profit?

Calculate \(P(|Y-\lambda| \leq 2 \sigma\) for the Poisson probability distribution of Example 3.22. Does this agree with the empirical rule? Equation transcription: Text transcription: P(|Y-\lambda| \leq 2 \sigma

Problem 214SE For simplicity, let us assume that there are two kinds of drivers. The safe drivers, who are 70% of the population, have probability .1 of causing an accident in a year. The rest of the population are accident makers, who have probability .5 of causing an accident in a year. The insurance premium is $400 times one’s probability of causing an accident in the following year. A new subscriber has an accident during the first year. What should be his insurance premium for the next year?

Problem 213SE A lot of N = 100 industrial products contains 40defectives. Let Y be the number of defectives in a random sample of size 20. Find p(10) by using (a) the hypergeometric probability distribution and (b) the binomial probability distribution. Is N large enough that the value for p(10) obtained from the binomial distribution is a good approximation to that obtained using the hypergeometric distribution?

Show that the hypergeometric probability function approaches the binomial in the limit as \(N \rightarrow \infty\) and \(p=r / N\) remains constant. That is, show that \(\lim _{N \rightarrow \infty} \frac{(r y) * N-r n-y)}{(N n)}=(n y) p^{y} q^{n-y}\) for \(p=r / N\) constant. Equation transcription: Text transcription: N rightarrow infty lim {N rightarrow infty} \frac{(r y) * N-r n-y)}{(N n)}=(n y) p^{y} q^{n-y} p=r / N

Problem 215SE It is known that 5% of the members of a population have disease A, which can be discovered by a blood test. Suppose that N (a large number) people are to be tested. This can be done in two ways: (1) Each person is tested separately, or (2) the blood samples of k people are pooled together and analyzed. (Assume that N = nk, with n an integer.) If the test is negative, all of them are healthy (that is, just this one test is needed). If the test is positive, each of the k persons must be tested separately (that is, a total of k + 1 tests are needed). a For fixed k, what is the expected number of tests needed in option 2? b Find the k that will minimize the expected number of tests in option 2. c If k is selected as in part (b), on the average how many tests does option 2 save in comparison with option 1?

Use the result derived in Exercise 3.216(c) and Definition 3.4 to derive directly the mean of a hypergeometric random variable.

Let have a hypergeometric distribution \(p(y)=\frac{(r y)(N-r n-y)}{(N n)}, y=0,1,2, \ldots, n\) a Show that \(P(Y=n)=p(n)=\left(\frac{r}{N}\right)\left(\frac{r-1}{N-1}\right)\left(\frac{r-2}{N-2}\right) \cdots\left(\frac{r-n+1}{N-n+1}\right)\) Write \(p(y) \) as \(p(y \mid r)\). Show that if \(r_{1}<r_{2}\), then \(\frac{p\left(y \mid r_{1}\right)}{p\left(y \mid r_{2}\right)}>\frac{p\left(y+1 \mid r_{1}\right)}{p\left(y+1 \mid r_{2}\right)}\) c Apply the binomial expansion to each factor in the following equation: \((1+a)^{N_{1}}(1+a)^{N_{2}}=(1+a)^{N_{1}+N_{2}}\) Now compare the coefficients of \(a^{n}\) on both sides to prove that \(\left(N_{1} 0\right)\left(N_{2} n\right)+\left(N_{1} 1\right)\left(N_{2} n-1\right)+\cdots+\left(N_{1} n\right)\left(N_{2} 0\right)=\left(N_{1}+N_{2} n\right)\) d Using the result of part (c), conclude that \(\sum_{y=0}^{n} p(y)=1\) Equation Transcription: Text Transcription: p(y)=(r y)(N-r n-y)/(N n), y=0,1,2,...,n. P(Y=n)=p(n)=(r/N)(r-1/N-1)(r-2/N-2)(r-n+1/N-n+1) p(y) p(y|r) r_1<r_2 p(y|r_1)/p(y|r_2)>p(y+1|r_1)/p(y+1|r_2) (1+a)^N_1(1+a)^N_2=(1+a)^N_1+N_2 (N_1 0) (N_2 n) +(N_1 1) (N_2 n-1) +...+(N_1 n) (N_2 0) =(N_1+N_2 n) sum_y=0^n p(y)=1

Problem 170E The U.S. mint produces dimes with an average diameter of .5 inch and standard deviation .01. Using Tchebysheff’s theorem, find a lower bound for the number of coins in a lot of 400 coins that are expected to have a diameter between .48 and .52.