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Probability And Statistical Inference 9th Edition Robert V. Hogg, Elliot Tanis, Dale Zimmerman - Solutions

Let X1 and X2 be a random sample of size n = 2 from the exponential distribution with pdf f(x) = 2e−2x, 0 < x < ∞. Find (a) P(0.5 < X1 < 1.0, 0.7 < X2 < 1.2). (b) E[X1(X2 − 0.5)2].
Let X1 and X2 be a random sample of size n = 2 from a distribution with pdf f(x) = 6x(1 − x), 0 < x < 1. Find the mean and the variance of Y = X1 + X2.
Suppose two independent claims are made on two insured homes, where each claim has pdf
Let X equal the outcome when a fair four-sided die that has its faces numbered 0, 1, 2, and 3 is rolled. Let Y equal the outcome when a fair four-sided die that has its faces numbered 0, 4, 8, and 12 is rolled. (a) Define the mgf of X. (b) Define the mgf of Y. (c) Let W = X + Y, the sum when the
The number of accidents in a period of one week follows a Poisson distribution with mean 2. The numbers of accidents from week to week are independent. What is the probability of exactly seven accidents in a given three weeks?
The number X of sick days taken during a year by an employee follows a Poisson distribution with mean 2. Let us observe four such employees. Assuming independence, compute the probability that their total number of sick days exceeds 10.
The number of cracks on a highway averages 0.5 per mile and follows a Poisson distribution. Assuming independence, what is the probability that, in a 40-mile stretch of that highway, there are fewer than 15 cracks?
Let X1 and X2 have independent distributions b(n1, p) and b(n2, p). Find the mgf of Y = X1 + X2. How is Y distributed?
The time X in minutes of a visit to a cardiovascular disease specialist by a patient is modeled by a gamma pdf with α = 1.5 and θ = 10. Suppose that you are such a patient and have four patients ahead of you. Assuming independence, what integral gives the probability that you will wait more than
Let X1 and X2 be two independent random variables. Let X1 and Y = X1 + X2 be χ2(r1) and χ2(r), respectively, where r1 < r. (a) Find the mgf of X2. (b) What is its distribution?
Generalize Exercise 5.4-3 by showing that the sum of n independent Poisson random variables with respective means μ1, μ2, . . . , μn is Poisson with mean μ1 + μ2 + · · · + μn.
Let X1, X2, X3, X4, X5 be a random sample of size 5 from a geometric distribution with p = 1/3. (a) Find the mgf of Y = X1 + X2 + X3 + X4 + X5. (b) How is Y distributed?
Let W = X1 + X2 + · · · + Xh, a sum of h mutually independent and identically distributed exponential random variables with mean θ. Show that W has a gamma distribution with parameters α = h and θ, respectively
A consumer buys n light bulbs, each of which has a lifetime that has a mean of 800 hours, a standard deviation of 100 hours, and a normal distribution. A light bulb is replaced by another as soon as it turns out. Assuming independence of the lifetimes, find the smallest n so that the succession of
Let the independent random variables X1 and X2 be N(0,1) and χ2(r), respectively. Let Y1 = X1/√X2/r and Y2 = X2. (a) Find the joint pdf of Y1 and Y2. (b) Determine the marginal pdf of Y1 and show that Y1 has a t distribution. (This is another, equivalent, way of finding the pdf of T.)
Let T have a t distribution with r degrees of freedom. Show that E(T) = 0 provided that r ≥ 2, and Var(T) = r/(r − 2) provided that r ≥ 3, by first finding E(Z), E(1/√U), E(Z2), and E(1/U).
Let n = 9 in the T statistic defined in Equation 5.5-2. (a) Find t0.025 so that P(−t0.025 ≤ T ≤ t0.025) = 0.95. (b) Solve the inequality [−t0.025 ≤ T ≤ t0.025] so that μ is in the middle.
5 . 5 - 2 . Let X be N(50, 36). Using the same set of axes, sketch the graphs of the probability density functions of(a) X.(b) , the mean of a random sample of size 9 from this distribution.(c) , the mean of a random sample of size 36 from this distribution.
Let X equal the weight of the soap in a “6-pound” box. Assume that the distribution of X is N(6.05, 0.0004). (a) Find P(X < 6.0171). (b) If nine boxes of soap are selected at random from the production line, find the probability that at most two boxes weigh less than 6.0171 pounds each. (c)
At a heat-treating company, iron castings and steel forgings are heat-treated to achieve desired mechanical properties and machinability. One steel forging is annealed to soften the part for each machining. Two lots of this part, made of 1020 steel, are heat-treated in two different furnaces. The
Let X denote the wing length in millimeters of a male gallinule and Y the wing length in millimeters of a female gallinule. Assume that X is N(184.09,39.37) and Y is N(171.93,50.88) and that X and Y are independent. If a male and a female gallinule are captured, what is the probability that X is
Let X and Y equal the respective numbers of hours a randomly selected child watches movies or cartoons on TV during a certain month. From experience, it is known that E(X) = 30, E(Y) = 50, Var(X) = 52, Var(Y) = 64, and Cov(X, Y) = 14. Twenty-five children are selected at random. Let Z equal the
At certain times during the year, a bus company runs a special van holding 10 passengers from Iowa City to Chicago. After the opening of sales of the tickets, the time (in minutes) between sales of tickets for the trip has a gamma distribution with α = 3 and θ = 2. (a) Assuming independence,
Suppose that the sick leave taken by the typical worker per year has μ = 10, σ = 2, measured in days. A firm has n = 20 employees. Assuming independence, how many sick days should the firm budget if the financial officer wants the probability of exceeding the number of days budgeted to be less
Let Y = X1 + X2 + · · · + X15 be the sum of a random sample of size 15 from the distribution whose pdf is f(x) = (3/2)X2, −1 < x < 1. Using the pdf of Y, we find that P(−0.3 ≤ Y ≤ 1.5) = 0.22788. Use the central limit theorem to approximate this probability
Approximate P(39.75 ≤ X ≤ 41.25), where X is the mean of a random sample of size 32 from a distribution with mean μ = 40 and variance σ2 = 8.
A random sample of size n = 18 is taken from the distribution with pdf f(x) = 1 − x/2, 0 ≤ x ≤ 2. (a) Find μ and σ2. (b) Find P(2/3 ≤ X ≤ 5/6), approximately.
Let X equal the weight in grams of a miniature candy bar. Assume that μ = E(X) = 24.43 and σ2 = Var(X) = 2.20. Let X be the sample mean of a random sample of n = 30 candy bars. Find (a) E(X). (b) Var(X). (c) P(24.17 ≤ X ≤ 24.82), Approximately.
In the casino game roulette, the probability of winning with a bet on red is p = 18/38. Let Y equal the number of winning bets out of 1000 independent bets that are placed. Find P(Y > 500), approximately.
If X is b(100,0.1), find the approximate value of P(12 ≤ X ≤ 14), using (a) The normal approximation. (b) The Poisson approximation. (c) The binomial.
A die is rolled 24 independent times. Let Y be the sum of the 24 resulting values. 0Recalling that Y is a random variable of the discrete type, approximate (a) P(Y ≥ 86). (b) P(Y < 86). (c) P(70 < Y ≤ 86).
Let Y equal the sum of n = 100 Bernoulli trials. That is, Y is b(100, p). For each of (i) p = 0.1, (ii) p = 0.5, and (iii) p = 0.8, (a) Draw the approximating normal pdfs, all on the same graph. (b) Find P(| Y/100 − p | ≤ 0.015), approximately.
Assume that the background noise X of a digital signal has a normal distribution with μ = 0 volts and σ = 0.5 volt. If we observe n = 100 independent measurements of this noise, what is the probability that at least 7 of them exceed 0.98 in absolute value? (a) Use the Poisson distribution to
Suppose that among gifted seventh-graders who score very high on a mathematics exam, approximately 20% are left-handed or ambidextrous. Let X equal the number of left-handed or ambidextrous students among a random sample of n = 25 gifted seventh-graders. Find P(2 < X < 9)(a) Using Table II in
Let X equal the number out of n = 48 mature aster seeds that will germinate when p = 0.75 is the probability that a particular seed germinates. Approximate P(35 ≤ X ≤ 40).
In adults, the pneumococcus bacterium causes 70% of pneumonia cases. In a random sample of n = 84 adults who have pneumonia, let X equal the number whose pneumonia was caused by the pneumococcus bacterium. Use the normal distribution to find P(X ≤ 52), approximately.
A candy maker produces mints that have a label weight of 20.4 grams. Assume that the distribution of the weights of these mints is N(21.37, 0.16). (a) Let denote the weight of a single mint selected at random from the production line. Find P( < 20.857). (b) During a particular shift, 100 mints
If E(X) = 17 and E(X2) = 298, use Chebyshev’s inequality to determine (a) A lower bound for P(10 < X < 24). (b) An upper bound for P(|X − 17| ≥ 16).
If the distribution of Y is b(n, 0.5), give a lower bound for P(|Y/n − 0.5| < 0.08) when (a) n = 100. (b) n = 500. (c) n = 1000.
Let be the mean of a random sample of size n = 15 from a distribution with mean μ = 80 and variance σ2 = 60. Use Chebyshev’s inequality to find a lower bound for P(75 < < 85).
The probability that a certain type of inoculation takes effect is 0.995. Use the Poisson distribution to approximate the probability that at most 2 out of 400 people given the inoculation find that it has not taken effect.
Let Y be χ2(n). Use the central limit theorem to demonstrate that W = (Y − n)/√2n has a limiting cdf that is N(0, 1).
In 1985, Kent Hrbek of the Minnesota Twins and Dion James of the Milwaukee Brewers had the following numbers of hits (H) and official at bats (AB) on grass and artificial turf:(a) Find the batting average BA (namely, H/AB) of each player on grass. (b) Find the BA of each player on artificial
Ledolter and Hogg report that a manufacturer of metal alloys is concerned about customer complaints regarding the lack of uniformity in the melting points of one of the firm€™s alloy filaments. Fifty filaments are selected and their melting points determined. The following results were
6 .1 - 6 . An insurance company experienced the following mobile home losses in 10,000€™s of dollars for 50 catastrophic events:(a) Using class boundaries 0.5, 5.5, 17.5, 38.5, 163.5, and 549.5, group these data into five classes. (b) Construct a relative frequency histogram of the
A small part for an automobile rearview mirror was produced on two different punch presses. In order to describe the distribution of the weights of those parts, a random sample was selected, and each piece was weighed in grams, resulting in the following data set:(a) Using about 10 (say, 8 to 12)
In Exercise 6.1-7, lead concentrations near the San Diego Freeway in 1976 are given. During the fall of 1977, the weekday afternoon lead concentrations (in Î¼g/m3) at the measurement station near the San Diego Freeway in Los Angeles were as follows:(a) Construct a frequency distribution and
When you purchase €œ1-pound bags€ of carrots, you can buy either €œbaby€ carrots or regular carrots. We shall compare the weights of 75 bags of each of these types of carrots. The following table gives the weights of the bags of baby carrots:This table
An insurance company experienced the following mobile home losses in 10,000€™s of dollars for 50 catastrophic events:(a) Find the five-number summary of the data and draw a box-and-whisker diagram.(b) Calculate the IQR and the locations of the inner and outer fences.(c) Draw a box plot that
In the casino game roulette, if a player bets $1 on red (or on black or on odd or on even), the probability of winning $1 is 18/38 and the probability of losing $1 is 20/38. Suppose that a player begins with $5 and makes successive $1 bets. Let Y equal the player€™s maximum capital before
The weights (in grams) of 25 indicator housings used on gauges are as follows:(a) Construct an ordered stem-and-leaf display, using integers as the stems and tenths as the leaves.(b) Find the five-number summary of the data and draw a box plot.(c) Are there any suspected outliers? Are there any
Nine measurements are taken on the strength of a certain metal. In order, they are 7.2, 8.9, 9.7, 10.5, 10.9, 11.7, 12.9, 13.9, 15.3, and these values correspond to the 10th, 20th, . . . , 90th percentiles of this sample. Construct a q–q plot of the measurements against the same percentiles of
An interior automotive supplier places several electrical wires in a harness. A pull test measures the force required to pull spliced wires apart. A customer requires that each wire that is spliced into the harness withstand a pull force of 20 pounds. Let X equal the pull force required to pull a
Let X equal the forced vital capacity (the volume of air a person can expel from his or her lungs) of a male freshman. Seventeen observations of X, which have been ordered, are(a) Find the median, the first quartile, and the third quartile.(b) Find the 35th and 65th percentiles.
In the expression for gr(y) = G€™r(y) in Equation 6.3-1, let n = 6, and r = 3, and write out the summations, showing that the €œtelescoping€ suggested in the text is achieved.In equation 6.3-1
Let W1 < W2 < · · · < Wn be the order statistics of n independent observations from a U(0, 1) distribution.(a) Find the pdf of W1 and that of Wn.(b) Use the results of (a) to verify that E(W1) = 1/(n + 1) and E(Wn) = n/(n + 1).(c) Show that the pdf of Wr is beta.
Let W1 < W2 < ··· < Wn be the order statistics of n independent observations from a U(0, 1) distribution.(a) Show that E(Wr2) = r(r + 1)/(n + 1)(n + 2), using a technique similar to that used in determining that E(Wr) = r/(n + 1).(b) Find the variance of Wr.
Let X1, X2, . . . , Xn be a random sample of size n from a geometric distribution for which p is the probability of success.(a) Use the method of moments to find a point estimate for p.(b) Explain intuitively why your estimate makes good sense.(c) Use the following data to give a point estimate of
Let X1, X2, . . . , Xn be a random sample from b(1, p) (i.e., n Bernoulli trials). Thus,(a) Show that = Y/n is an unbiased estimator of p.(b) Show that Var(X) = p(1 ˆ’ p)/n.(c) Show that E[(1 ˆ’ X)/n] = (n ˆ’ 1)[p(1 ˆ’ p)/n2].(d) Find the value of c so that c(1 ˆ’ ) is an unbiased
Let X1, X2, . . . , Xn be a random sample of size n from a normal distribution.(a) Show that an unbiased estimator of Ïƒ is cS, where(b) Find the value of c when n = 5; when n = 6.(c) Graph c as a function of n. What is the limit of c as n increases without bound?
An urn contains 64 balls, of which N1 are orange and N2 are blue. A random sample of n = 8 balls is selected from the urn without replacement, and X is equal to the number of orange balls in the sample. This experiment was repeated 30 times (the 8 balls being returned to the urn before each
Let independent random samples, each of size n, be taken from the k normal distributions with means μj = c + d[j − (k + 1)/2], j = 1, 2, . . . , k, respectively, and common variance σ2. Find the maximum likelihood estimators of c and d.
A random sample X1, X2, ... , Xn of size n is taken from N( Î¼, Ïƒ2), where the variance Î¸ = Ïƒ2 is such that 0and that this estimator is an unbiased estimator of Î¸.
For determining half-lives of radioactive isotopes, it is important to know what the background radiation is in a given detector over a specific period. The following data were taken in a Î³-ray detection experiment over 98 ten-second intervals:Assume that these data are observations of a Poisson
Let(a) Show that the maximum likelihood estimator of Î¸ is
The “golden ratio” is φ = (1 + √5)/2. John Putz, a mathematician who was interested in music, analyzed Mozart’s sonata movements, which are divided into two distinct sections, both of which are repeated in performance (see References). The length of the “Exposition” in measures is
In some situations where the regression model is useful, it is known that the mean of Y when X = 0 is equal to 0, i.e., Yi = Î²xi + Îµi where Îµi for i = 1, 2, . . . , n are independent and N(0, Ïƒ2).
The final grade in a calculus course was predicted on the basis of the student€™s high school grade point average in mathematics, Scholastic Aptitude Test (SAT) score in mathematics, and score on a mathematics entrance examination. The predicted grades x and the earned grades y for 10 students
Let x and y equal the ACT scores in social science and natural science, respectively, for a student who is applying for admission to a small liberal arts college. A sample of n = 15 such students yielded the following data:(a) Calculate the least squares regression line for these data.(b) Plot the
The data in the following table, part of a set of data collected by Ledolter and Hogg, provide the number of miles per gallon (mpg) for city and highway driving of 2007 midsize-model cars, as well as the curb weight of the cars:(a) Find the least squares regression line for highway mpg (y) and city
Let X1, X2, . . . , Xn denote a random sample from b(1, p). We know that x̄ is an unbiased estimator of p and that Var(x̄) = p(1 − p)/n.(a) Find the Rao–Cramér lower bound for the variance of every unbiased estimator of p.(b) What is the efficiency of X as an estimator of p?
Find the Rao–Cramér lower bound, and thus the asymptotic variance of the maximum likelihood estimator θ, if the random sample X1, X2, . . . , Xn is taken from each of the distributions having the following pdfs: (a) f(x; θ) = (1/θ2) x e−x/θ, 0 < x < ∞, 0 < θ < ∞. (b) f(x; θ) =
Find a sufficient statistic for Î¸, given a random sample, X1, X2, . . . , Xn, from a distribution with pdf
Let X1, X2, . . . , Xn be a random sample from N(0, σ2), where n is odd. Let Y and Z be the mean and median of the sample. Argue that Y and Z − Y are independent so that the variance of Z is Var(Y) + Var(Z− Y). We know that Var(Y) = σ2/n, so that we could estimate the Var(Z − Y) by Monte
Let X1, X2, . . . , Xn be a random sample from a Poisson distribution with mean λ > 0. Find the conditional probability P(X1 = x1, . . . , Xn = xn | Y = y), where Y = X1 + · · · + Xn and the nonnegative integers x1, x2, . . . , xn sum to y. Note that this probability does not depend on λ.
Let X1, X2, . . . , Xn be a random sample from a distribution with pdf f(x; Î¸) = Î¸xÎ¸ˆ’1, 0 (a) Find a sufficient statistic Y for Î¸.
Let X1, X2, . . . , Xn be a random sample from a gamma distribution with known parameter Î± and unknown parameter Î¸ > 0(b) Show that the maximum likelihood estimator of Î¸ is a function of Y and is an unbiased estimator of Î¸.
Let X1, X2, . . . , Xn be a random sample from N(0, Î¸), where Ïƒ2 = Î¸ > 0 is unknown. Argue that the sufficient statisticare independent.
Let X1, X2, . . . , Xn be a random sample from a gamma distribution with known α and with θ = 1/τ. Say τ has a prior pdf that is gamma with parameters α0 and θ0, so that the prior mean is α0θ0.(a) Find the posterior pdf of τ, given that X1 = x1, X2 = x2, . . . , Xn = xn.(b) Find the mean
Consider a random sample X1, X2, . . . , Xn from a distribution with pdfLet Î¸ have a prior pdf that is gamma with Î± = 4 and the usual Î¸ = 1/4. Find the conditional mean of Î¸, given that X1 = x1, X2 = x2, . . . , Xn = xn.
Let Y be the largest order statistic of a random sample of size n from a distribution with pdf f(x | Î¸) = 1/Î¸, 0where Î± > 0, Î² > 0.(a) If w(Y) is the Bayes estimator of Î¸ and [Î¸ ˆ’ w(Y)]2 is the loss function, find w(Y).(b) If n = 4, Î± = 1, and Î² = 2, find the
Consider the likelihood function L(α, β, σ2) of Section 6.5. Let α and β be independent with priors N(α1, σ12) and N(β0, σ02). Determine the posterior mean of α + β(x − x).
Suppose X is b(n, Î¸) and Î¸ is beta(Î±, Î²). Show that the marginal pdf of X (the compound distribution) isFor x = 0, 1, 2, . . . , n.
Let X have the pdf
Let X1, X2 be a random sample from the Cauchy distribution with pdfConsider the non-informative prior h(Î¸1, Î¸2) ˆ 1 on that support. Obtain the posterior pdf (except for constants) of Î¸1, Î¸2 if x1 = 3 and x2 = 7. For estimates, find Î¸1, Î¸2 that maximizes this posterior pdf;
A leakage test was conducted to determine the effectiveness of a seal designed to keep the inside of a plug airtight. An air needle was inserted into the plug, and the plug and needle were placed under water. The pressure was then increased until leakage was observed. Let equal the pressure in
In nuclear physics, detectors are often used to measure the energy of a particle. To calibrate a detector, particles of known energy are directed into it. The values of signals from 15 different detectors, for the same energy, are(a) Find a 95% confidence interval for Î¼, assuming that these are
Let X1, X2, ... , Xn be a random sample of size n from the normal distribution N(μ, σ2). Calculate the expected length of a 95% confidence interval for μ, assuming that n = 5 and the variance is(a) Known.(b) Unknown.
Let S2 be the variance of a random sample of size n from N(Î¼, Ïƒ2). Using the fact that (n ˆ’ 1)S2/Ïƒ2 is Ï‡2(nˆ’1), note that the probabilityWhere Rewrite the inequalities to obtain If n = 13 and Show that [6.11, 24.57] is a 90%
A random sample of size 8 from N(μ, 72) yielded = 85. Find the following confidence intervals for μ: (a) 99%. (b) 95%. (c) 90%. (d) 80%.
Let X equal the weight in grams of a €œ52-gram€ snack pack of candies. Assume that the distribution of X is N(Î¼, 4). A random sample of n = 10 observations of X yielded the following data:(a) Give a point estimate for Î¼.(b) Find the endpoints for a 95% confidence interval for Î¼.(c)
To determine whether the bacteria count was lower in the west basin of Lake Macatawa than in the east basin, n = 37 samples of water were taken from the west basin and the number of bacteria colonies in 100 milliliters of water was counted. The sample characteristics were = 11.95 and s = 11.80,
Assume that the yield per acre for a particular variety of soybeans is N(μ, σ2). For a random sample of n = 5 plots, the yields in bushels per acre were 37.4, 48.8, 46.9, 55.0, and 44.0.(a) Give a point estimate for μ.(b) Find a 90% confidence interval for μ.
Twenty-four 9th- and 10th-grade high school girls were put on an ultra-heavy rope-jumping program. The following data give the time difference for each girl (€œbefore program time€ minus €œafter program time€) for the 40-yard dash:(a) Give a point estimate of Î¼D, the mean of the
Let X and Y equal the hardness of the hot and cold water, respectively, in a campus building. Hardness is measured in terms of the calcium ion concentration (in ppm). The following data were collected (n = 12 observations of X and m = 10 observations of Y):(a) Calculate the sample means and the
Let S2X and S2Y be the respective variances of two independent random samples of sizes n and m from N(Î¼X, Ïƒ2X) and N(Î¼Y, Ïƒ2Y). Use the fact that F = [S2Y/Ïƒ2Y]/[S2X/Ïƒ2X] has an F distribution, with parameters r1 = mˆ’
Let X1, X2, . . . , X5 be a random sample of SAT mathematics scores, assumed to be N(Î¼X, Ïƒ2), and let Y1, Y2, . . . , Y8 be an independent random sample of SAT verbal scores, assumed to be N(Î¼Y, Ïƒ2). If the following data are observed, find a 90% confidence interval for Î¼X ˆ’
Let X and Y equal, respectively, the blood volumes in milliliters for a male who is a paraplegic and participates in vigorous physical activities and for a male who is able-bodied and participates in everyday, ordinary activities. Assume that X is N(Î¼X, Ïƒ2X) and Y is N(Î¼Y, Ïƒ2Y).
A test was conducted to determine whether a wedge on the end of a plug fitting designed to hold a seal onto the plug was doing its job. The data taken were in the form of measurements of the force required to remove a seal from the plug with the wedge in place (say, X) and the force required
Let X-bar, Y-bar, S2X, and S2Y be the respective sample means and unbiased estimates of the variances obtained from independent samples of sizes n and m from the normal distributions N(Î¼X, Ïƒ2X) and N(Î¼Y, Ïƒ2Y), where Î¼X,

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