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Introduction To Mathematical Statistics And Its Applications 5th Edition Richard J. Larsen, Morris L. Marx - Solutions

Fifty spotlights have just been installed in an outdoor security system. According to the manufacturer's specifications, these particular lights are expected to burn out at the rate of 1.1 per one hundred hours. What is the expected number of bulbs that will fail to last for at least seventy-five
Five hundred people are attending the first annual "I was Hit by Lighting" Club. Approximate the probability that at most one of the five hundred was born on Poisson's birthday.
Suppose you want to invent a new superstition that "Bad things come in fours." Using the data given in Case Study 4.2.4 and the type of analysis described on p. 238, calculate the probability that your superstition would appear to be true.
A chromosome mutation linked with colorblindness is known to occur, on the average, once in every ten thousand births.(a) Approximate the probability that exactly three of the next twenty thousand babies born will have the mutation.(b) How many babies out of the next twenty thousand would have to
Suppose that 1% of all items in a supermarket are not priced properly. A customer buys ten items. What is the probability that she will be delayed by the cashier because one or more of her items require a price check? Calculate both a binomial answer and a Poisson answer. Is the binomial model
A newly formed life insurance company has underwritten term policies on 120 women between the ages of forty and forty-four. Suppose that each woman has a 1/150 probability of dying during the next calendar year, and that each death requires the company to pay out $50,000 in benefits. Approximate
According to an airline industry report (178), roughly 1 piece of luggage out of every 200 that are checked is lost. Suppose that a frequent-flying businesswoman will be checking 120 bags over the course of the next year. Approximate the probability that she will lose 2 of more pieces of luggage.
Electromagnetic fields generated by power transmission lines are suspected by some researchers to be a cause of cancer. Especially at risk would be telephone linemen because of their frequent proximity to high-voltage wires. According to one study, two cases of a rare form of cancer were detected
Astronomers estimate that as many as one hundred billion stars in the Milky Way galaxy are encircled by planets. If so, we may have a plethora of cosmic neighbors. Let p denote the probability that any such solar system contains intelligent life. How small can p be and still give a fifty-fifty
State Tech's basketball team, the Fighting Logarithms, have a 70% foul-shooting percentage.(a) Write a formula for the exact probability that out of their next one hundred free throws, they will make between seventy-five and eighty, inclusive.(b) Approximate the probability asked for in part (a).
A random sample of 747 obituaries published recently in Salt Lake City newspapers revealed that 344 (or 46%) of the decedents died in the three-month period following their birthdays (123). Assess the statistical significance of that finding by approximating the probability that 46% or more would
There is a theory embraced by certain parapsychologists that hypnosis can enhance a person's ESP ability. To test that hypothesis, an experiment was set up with fifteen hypnotized subjects (21). Each was asked to make 100 guesses using the same sort of ESP cards and protocol that were described in
If pX(k) = (10/k)(0.7)k(0.3)10−k, k = 0, 1, . . . , 10, is it appropriate to approximate P(4 ≤ X ≤ 8) by computing the following?Explain.
A sell-out crowd of 42,200 is expected at Cleveland's Jacobs Field for next Tuesday's game against the Baltimore Orioles, the last before a long road trip. The ballpark's concession manager is trying to decide how much food to have on hand. Looking at records from games played earlier in the
A fair coin is tossed two hundred times. Let Xi = 1 if the ith toss comes up heads and Xi = 0 otherwise, i = 1, 2, . . . , 200; X = Calculate the central limit theorem approximation for P(|X − E(X)| ≤ 5). How does this differ from the DeMoivre-Laplace approximation?
Suppose that one hundred fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370. Include a continuity correction in your analysis.
Let X be the amount won or lost in betting $5 on red in roulette. Then px(5) = 18/38 and px(−5) = 20/38. If a gambler bets on red one hundred times, use the central limit theorem to estimate the probability that those wagers result in less than $50 in losses.
If X1, X2, . . . , Xn are independent Poisson random variables with parameters λ1,λ2, . . . , λn, respectively, and if X = X1 + X2 + ・・・ + Xn, then X is a Poisson random variable with parameter λ = . What specific form does the ratio in Theorem 4.3.2 take if the Xi's are Poisson random
An electronics firm receives, on the average, fifty orders per week for a particular silicon chip. If the company has sixty chips on hand, use the central limit theorem to approximate the probability that they will be unable to fill all their orders for the upcoming week. Assume that weekly demands
Let Z be a standard normal random variable. Use Appendix Table A.1 to find the numerical value for each of the following probabilities. Show each of your answers as an area under fZ(z). (a) P(0 ≤ Z ≤ 2.07) (b) P(−0.64 ≤ Z < −0.11) (c) P(Z > −1.06) (d) P(Z < −2.33) (e) P(Z ≥ 4.61)
Considerable controversy has arisen over the possible aftereffects of a nuclear weapons test conducted in Nevada in 1957. Included as part of the test were some three thousand military and civilian "observers." Now, more than fifty years later, eight cases of leukemia have been diagnosed among
Econo-Tire is planning an advertising campaign for its newest product, an inexpensive radial. Preliminary road tests conducted by the firm's quality-control department have suggested that the lifetimes of these tires will be normally distributed with an average of thirty thousand miles and a
A large computer chip manufacturing plant under construction in West bank is expected to result in an additional fourteen hundred children in the county's public school system once the permanent workforce arrives. Any child with an IQ under 80 or over 135 will require individualized instruction
Records for the past several years show that the amount of money collected daily by a prominent televangelist is normally distributed with a mean (μ) of $20,000 and a standard deviation (σ) of $5000. What are the chances that tomorrow's donations will exceed $30,000?
The following letter was written to a well-known dispenser of advice to the lovelorn (171): Dear Abby: You wrote in your column that a woman is pregnant for 266 days. Who said so? I carried my baby for ten months and five days, and there is no doubt about it because I know the exact date my baby
A criminologist has developed a questionnaire for predicting whether a teenager will become a delinquent. Scores on the questionnaire can range from 0 to 100, with higher values reflecting a presumably greater criminal tendency. As a rule of thumb, the criminologist decides to classify a teenager
The cross-sectional area of plastic tubing for use in pulmonary resuscitators is normally distributed withμ = 12.5 mm2 and σ = 0.2 mm2.When the area is less than 12.0 mm2 or greater than 13.0 mm2, the tube does not fit properly. If the tubes are shipped in boxes of one thousand, how many
At State University, the average score of the entering class on the verbal portion of the SAT is 565, with a standard deviation of 75. Marian scored a 660. How many of State's other 4250 freshmen did better? Assume that the scores are normally distributed.
A college professor teaches Chemistry 101 each fall to a large class of freshmen. For tests, she uses standardized exams that she knows from past experience produce bell-shaped grade distributions with a mean of 70 and a standard deviation of 12. Her philosophy of grading is to impose standards
Suppose the random variable Y can be described by a normal curve with μ=40. For what value of σ isP(20 ≤ Y ≤ 60) = 0.50
(a) Let 0 < a < b. Which number is larger?(b) Let a >0. Which number is larger?
It is estimated that 80% of all eighteen-year old women have weights ranging from 103.5 to 144.5 lb. Assuming the weight distribution can be adequately modeled by a normal curve and that 103.5 and 144.5 are equidistant from the average weight μ, calculate σ.
Recall the breath analyzer problem described in Example 4.3.5. Suppose the driver's blood alcohol concentration is actually 0.09% rather than 0.075%. What is the probability that the breath analyzer will make an error in his favor and indicate that he is not legally drunk? Suppose the police offer
If a random variable Y is normally distributed with mean μ and standard deviation σ, the Z ratio Y – μ/σ is often referred to as a normed score: It indicates the magnitude of y relative to the distribution from which it came. "Norming" is sometimes used as an affirmative action mechanism in
The IQs of nine randomly selected people are recorded. Let Y denote their average. Assuming the distribution from which the Yi's were drawn is normal with a mean of 100 and a standard deviation of 16, what is the probability that Y will exceed 103? What is the probability that any arbitrary Yi will
Let Y1, Y2, . . . , Yn be a random sample from a normal distribution where the mean is 2 and the variance is 4. How large must n be in order thatP(1.9 ≤ Y̅≤ 2.1) ≥ 0.99
A circuit contains three resistors wired in series. Each is rated at 6 ohms. Suppose, however, that the true resistance of each one is a normally distributed random variable with a mean of 6 ohms and a standard deviation of 0.3 ohm. What is the probability that the combined resistance will exceed
The cylinders and pistons for a certain internal combustion engine are manufactured by a process that gives a normal distribution of cylinder diameters with a mean of 41.5 cm and a standard deviation of 0.4 cm. Similarly, the distribution of piston diameters is normal with a mean of 40.5 cm and a
Use moment-generating functions to prove the two corollaries to Theorem 4.3.3.
Let Y1, Y2, . . . , Y9 be a random sample of size 9 from a normal distribution where μ = 2 and σ = 2. Let Y∗1, Y∗2, . . . , Y∗9 be an independent random sample from a normal distribution having μ = 1 and σ = 1. Find P(Y̅ ≥ Y̅ ∗).
(a) Evaluate(b) Evaluate
Assume that the random variable Z is described by a standard normal curve fZ(z). For what values of z are the following statements true?(a) P(Z ≤ z) = 0.33(b) P(Z ≥ z) = 0.2236(c) P(−1.00 ≤ Z ≤ z) = 0.5004(d) P(−z < Z < z) = 0.80(e) P(z ≤ Z ≤ 2.03) = 0.15
Let zα denote the value of Z for which P(Z ≥ zα) = α. By definition, the interquartile range, Q, for the standard normal curve is the difference Q = z.25 − z.75 Find Q.
Oak Hill has 74,806 registered automobiles. A city ordinance requires each to display a bumper decal showing that the owner paid an annual wheel tax of $50. By law, new decals need to be purchased during the month of the owner's birthday. This year's budget assumes that at least $306,000 in decal
Hertz Brothers, a small, family-owned radio manufacturer, produces electronic components domestically but subcontracts the cabinets to a foreign supplier. Although inexpensive, the foreign supplier has a quality control program that leaves much to be desired. On the average, only 80% of the
Fifty-five percent of the registered voters in Sheridanville favor their incumbent mayor in her bid for re-election. If four hundred voters go to the polls, approximate the probability that (a) The race ends in a tie. (b) The challenger scores an upset victory.
Because of her past convictions for mail fraud and forgery, Jody has a 30% chance each year of having her tax returns audited. What is the probability that she will escape detection for at least three years? Assume that she exaggerates, distorts, misrepresents, lies, and cheats every year.
The factorial moment-generating function for any random variable W is the expected value of tw. Moreover dr/dtr E(tW)|t=1 = E[W(W ˆ’ 1)ãƒ»ãƒ»ãƒ»(W ˆ’ r + 1)]. Find the factorial moment-generating function for a geometric random variable and use it to verify the expected value and
A teenager is trying to get a driver's license. Write out the formula for the pdf px(k), where the random variable X is the number of tries that he needs to pass the road test. Assume that his probability of passing the exam on any given attempt is 0.10. On the average, how many attempts is he
Is the following set of data likely to have come from the geometric pdf pX(k) = (3/4)k−1・(1/4), k = 1, 2, . . .? Explain.
Recently married, a young couple plans to continue having children until they have their first girl. Suppose the probability that a child is a girl is 1 2 , the outcome of each birth is an independent event, and the birth at which the first girl appears has a geometric distribution. What is the
Show that the cdf for a geometric random variable is given by FX(t) = P(X ≤ t) = 1 − (1 − p)[t], where [t] denotes the greatest integer in t, t ≥ 0.
Suppose three fair dice are tossed repeatedly. Let the random variable X denote the roll on which a sum of 4 appears for the first time. Use the expression for Fx(t) given in Question 4.4.5 to evaluate P(65≤ X ≤75).
Let Y be an exponential random variable, where fY(y) = λe−λy, 0 ≤ y. For any positive integer n, show that P(n ≤ Y ≤ n + 1) = e−λn(1 − e−λ). If p = 1 − e−λ, the "discrete" version of the exponential pdf is the geometric pdf.
Sometimes the geometric random variable is defined to be the number of trials, X, preceding the first success. Write down the corresponding pdf and derive the moment-generating function for X two ways-(1) by evaluating E(etX) directly and (2) by using Theorem 3.12.3.
Differentiate the moment-generating function for a geometric random variable and verify the expressions given for E(X) and Var(X) in Theorem 4.4.1.
A door-to-door encyclopedia salesperson is required to document five in-home visits each day. Suppose that she has a 30% chance of being invited into any given home, with each address representing an independent trial. What is the probability that she requires fewer than eight houses to achieve her
Suppose that X1, X2, . . . , Xk are independent negative binomial random variables with parameters r1 and p, r2 and p, . . ., and rk and p, respectively. Let X = X1 + X2+・・ + Xk. Find MX(t), pX(t), E(X), and Var(X).
An underground military installation is fortified to the extent that it can withstand up to three direct hits from air-to-surface missiles and still function. Suppose an enemy aircraft is armed with missiles, each having a 30% chance of scoring a direct hit. What is the probability that the
Darryl's statistics homework last night was to flip a fair coin and record the toss, X, when heads appeared for the second time. The experiment was to be repeated a total of one hundred times. The following are the one hundred values for X that Darryl turned in this morning. Do you think that he
When a machine is improperly adjusted, it has probability 0.15 of producing a defective item. Each day, the machine is run until three defective items are produced. When this occurs, it is stopped and checked for adjustment. What is the probability that an improperly adjusted machine will produce
For a negative binomial random variable whose pdf is given by Equation 4.5.1, find E(X) directly by evaluating pr(1 − p)k−r.
Let the random variable X denote the number of trials in excess of r that are required to achieve the rth success in a series of independent trials, where p is the probability of success at any given trial. Show that
Calculate the mean, variance, and moment generating function for a negative binomial random variable X whose pdf is given by the expression
Let X1, X2, and X3 be three independent negative binomial random variables with pdfsfor i = 1, 2, 3. Define X = X1 + X2 + X3. Find P(10 ≤ X ≤ 12). (Use the moment-generating functions of X1, X2, and X3 to deduce the pdf of X.)
Differentiate the moment-generating function MX(t) = [pet/1−(1−p)et]r to verify the formula given in Theorem 4.5.1 for E(X).
An Arctic weather station has three electronic wind gauges. Only one is used at any given time. The lifetime of each gauge is exponentially distributed with a mean of one thousand hours. What is the pdf of Y, the random variable measuring the time until the last gauge wears out?
Differentiate the gamma moment-generating function to show that the formula for E(Ym) given in Question 4.6.8 holds for arbitrary r > 0.
A service contact on a new university computer system provides twenty-four free repair calls from a technician. Suppose the technician is required, on the average, three times a month. What is the average time it will take for the service contract to be fulfilled?
Suppose a set of measurements Y1, Y2, . . . , Y100 is taken from a gamma pdf for which E(Y) = 1.5 and Var(Y) = 0.75. How many Yi's would you expect to find in the interval [1.0, 2.5]?
Demonstrate that λ plays the role of a scale parameter by showing that if Y is gamma with parameters r and λ, then λY is gamma with parameters r and 1.
Show that a gamma pdf has the unique mode r − 1/λ; that is, show that the function fY(y) = λr/ Г(r)yr−1e−λy takes its maximum value at ymode = r−1/λ and at no other point.
Prove that Г(1/2) = √π. [Consider E(Z2), where Z is a standard normal random variable.]
Show that Г(7/2) = 15/8 √π.
If the random variable Y has the gamma pdf with integer parameter r and arbitrary λ > 0, show that E(Ym) = (m + r −1)!/(r −1)!λm
Differentiate the gamma moment-generating function to verify the formulas for E(Y) and Var(Y) given in Theorem 4.6.3.
A random sample of size 8 – X1 = 1, X2 = 0, X3 = 1, X4 = 1, X5 = 0, X6 = 1, X7 = 1, and X8 = 0 – is taken from the probability function pX(k; θ) = θk(1 − θ)1−k, k = 0, 1; 0 < θ < 1. Find the maximum likelihood estimate for θ.
(a) Based on the random sample Y1 = 6.3, Y2 = 1.8, Y3 = 14.2, and Y4 = 7.6, use the method f maximum likelihood to estimate the parameter θ in the uniform pdf fY(y; θ) = 1/θ, 0 ≤ y ≤ θ(b) Suppose the random sample in part (a) represents the two-parameter uniform pdf fY(y; θ1, θ2) =
Find the maximum likelihood estimate for θ in the pdf fY(y; θ) = 2y/1 − θ2, θ ≤ y ≤ 1if a random sample of size 6 yielded the measurements 0.70, 0.63, 0.92, 0.86, 0.43, and 0.21.
A random sample of size n is taken from the pdf fY(y; θ) = 2y/θ2, 0 ≤ y ≤ θ Find an expression for ˆθ, the maximum likelihood estimator for θ.
If the random variable Y denotes an individual's income, Pareto's law claims that P(Y ≥ y) = (k/y)θ, where k is the entire population's minimum income. It follows that FY(y) = 1− (k/y)θ, and, by differentiation,fY(y; θ) = θkθ(1/y)θ+1, y ≥ k; θ ≥ 1Assume k is known. Find the maximum
The exponential pdf is a measure of lifetimes of devices that do not age. However, the exponential pdf is a special case of the Weibull distribution, which measures time to failure of devices where the probability of failure increases as time does. A Weibull random variable Y has pdf fY(y; α,β) =
Suppose a random sample of size n is drawn from a normal pdf where the mean μ is known but the variance σ2 is unknown. Use the method of maximum likelihood to find a formula for ˆσ2. Compare your answer to the maximum likelihood estimator found in Example 5.2.4.
Let y1, y2, . . . , yn be a random sample of size n from the pdf fY(y; θ) = 2y/θ2, 0 ≤ y ≤ θ. Find a formula for the method of moments estimate for θ. Compare the values of the method of moments estimate and the maximum likelihood estimate if a random sample of size 5 consists of the
Use the method of moments to estimate θ in the pdffY(y; θ) = (θ2 + θ)yθ−1(1 − y), 0 ≤ y ≤ 1Assume that a random sample of size n has been collected.
Find the method of moments estimate for λ if a random sample of size n is taken from the exponential pdf,fY(y; λ) = λe−λy, y ≥ 0.
The number of red chips and white chips in an urn is unknown, but the proportion, p, of reds is either 1/3 or 1/2. A sample of size 5, drawn with replacement, yields the sequence red, white, white, red, and white. What is the maximum likelihood estimate for p?
Suppose that Y1 = 8.3, Y2 = 4.9, Y3 = 2.6, and Y4 = 6.5 is a random sample of size 4 from the two-parameter uniform pdf,fY(y; θ1, θ2) = 1/2θ2, θ1 − θ2 ≤ y ≤ θ1 + θ2Use the method of moments to calculate θ1e and θ2e.
Find a formula for the method of moments estimate for the parameter θ in the Pareto pdf,fY(y; θ) = θkθ (1/y)θ+1, y ≥ k; θ ≥ 1Assume that k is known and that the data consist of a random sample of size n. Compare your answer to the maximum likelihood estimator found in Question 5.2.13.
Calculate the method of moments estimate for the parameter θ in the probability functionpX(k; θ) = θk(1 − θ)1−k, k = 0, 1if a sample of size 5 is the set of numbers 0, 0, 1, 0, 1.
Find the method of moments estimates for μ and σ2, based on a random sample of size n drawn from a normal pdf, where μ= E(Y) and σ2 = Var(Y). Compare your answers with the maximum likelihood estimates derived in Example 5.2.4.
Use the method of moments to derive formulas for estimating the parameters r and p in the negative binomial pdf,
Bird songs can be characterized by the number of clusters of "syllables" that are strung together in rapid succession. If the last cluster is defined as a "success," it may be reasonable to treat the number of clusters in a song as a geometric random variable. Does the model pX(k) = (1 − p)k−1
Let y1, y2, . . . , yn be a random sample from the continuous pdf fY(y; θ1, θ2). Let ˆσ2 = 1/n Show that the solutions of the equations E(Y) = y and Var(Y)= ˆσ2 for θ1 and θ2 give the same results as using the equations in Definition 5.2.3.
Use the sample Y1 = 8.2, Y2 = 9.1, Y3 = 10.6, and Y4 = 4.9 to calculate the maximum likelihood estimate for λ in the exponential pdffY(y; λ) = λe−λy, y ≥ 0
Suppose a random sample of size n is drawn from the probability modelpX(k; θ) = θ2ke−θ2/k! , k = 0, 1, 2, . . .Find a formula for the maximum likelihood estimator, ˆθ.
Given that Y1 = 2.3, Y2 = 1.9, and Y3 = 4.6 is a random sample fromfY(y; θ) = y3e−y/θ/6θ4, y ≥ 0calculate the maximum likelihood estimate for θ.
Use the method of maximum likelihood to estimate θ in the pdffY(y; θ) = θ/2√y e−θ√y, y ≥ 0Evaluate θe for the following random sample of size 4: Y1 = 6.2, Y2 = 7.0, Y3 = 2.5, and Y4 = 4.2.
An engineer is creating a project scheduling program and recognizes that the tasks making up the project are not always completed on time. However, the completion proportion tends to be fairly high. To reflect this condition, he uses the pdffY(y; θ) = θ yθ−1, 0 ≤ y ≤ 1, and 0 < θwhere
The following data show the number of occupants in passenger cars observed during one hour at a busy intersection in Los Angeles (69). Suppose it can be assumed that these data follow a geometric distribution, pX(k; p) = (1 − p)k−1 p, k = 1, 2, . . .. Estimate p and compare the observed and

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