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John E Freunds Mathematical Statistics With Applications 8th Edition Irwin Miller, Marylees Miller - Solutions

Show that for the unbiased estimator of Example 10.4, n + 1 / n ∙ Yn, the Cramer-Rao inequality is not satisfied.
The information about θ in a random sample of size n is also given byWhere f (x) is the value of the population density at x, provided that the extremes of the region for which f(x) ≠ 0 do not depend on θ. The derivation of this formula takes the following steps: (a) Differentiating the
Rework Example 10.5 using the alternative formula for the information given in Exercise 10.19. Example 10.5 Show that is a minimum variance unbiased estimator of the mean µ of a normal population.
If X̅1 is the mean of a random sample of size n from a normal population with the mean µ and the variance σ21, X̅2 is the mean of a random sample of size n from a normal population with the mean µ and the variance σ22, and the two samples are independent, show that(a) ω ∙ X̅1 +(1 – ω)
With reference to Exercise 10.21, find the efficiency of the estimator of part (a) with Ï‰ = 1/2 relative to this estimator withIn exercise If 1 is the mean of a random sample of size n from a normal population with the mean µ and the variance Ïƒ21, 2 is the mean of a
If X̅1 and X̅2 are the means of independent random samples of sizes n1 and n2 from a normal population with the mean µ and the variance σ2, show that the variance of the unbiased estimator Is a minimum when ω = n1 / n1 + n2
With reference to Exercise 10.23, find the efficiency of the estimator with ω = 1/2 relative to the estimator with ω = n1/ n1 + n2.
If X1, X2, and X3 constitute a random sample of size n = 3 from a normal population with the mean µ and the variance σ2, find the efficiency of X1 + 2X2 + X3 / 4 relative to X1 + X2 + X3 / 3 as estimates of µ.
If X1 and X2 constitute a random sample of size n = 2 from an exponential population, find the efficiency of 2Y1 relative to X̅, where Y1 is the first order statistic and 2Y1 and X̅ are both unbiased estimators of the parameter θ.
Verify the result given for var(n + 1 / n ∙ Yn) in Example 10.6.
With reference to Example 10.3, we showed on page 281 that X̅ – 1 is an unbiased estimator of d, and in Exercise 10.8 the reader was asked to find another unbiased estimator of d based on the smallest sample value. Find the efficiency of the first of these two estimators relative to the
With reference to Exercise 10.12, show that 2X – 1 is also an unbiased estimator of k, and find the efficiency of this estimator relative to the one of part (b) of Exercise 10.12 for (a) n = 2; (b) n = 3.
Use the formula for the sampling distribution of 8 X on page 253 to show that for random samples of size n = 3 the median is an unbiased estimator of the parameter θ of a uniform population with α = θ – 12 and β = θ + 12.
Since the variances of the mean and the midrange are not affected if the same constant is added to each observation, we can determine these variances for random samples of size 3 from the uniform populationBy referring instead to the uniform population(a) Show that E(X) = 1/2 , E(X2) = 1/3 , and
Show that if Θ is a biased estimator of θ, then
If Θ1 = X/n , Θ2 = X + 1 / n+ 2 , and Θ3 = 1/3 are estimators of the parameter θ of a binomial population and θ = 1/2 , for what values of n is (a) The mean square error of Θ2 less than the variance of Θ1; (b) The mean square error of Θ3 less than the variance of Θ1?
Use Definition 10.5 to show that Y1, the first order statistic, is a consistent estimator of the parameter Î± of a uniform population with Î² = Î± + 1.Definition 10.5The statistic is a consistent estimator of the parameter of a given distribution if and only if
With reference to Exercise 10.33, use Theorem 10.3 to show that Y1 – 1/n+1 is a consistent estimator of the parameter α.
With reference to the uniform population of Example 10.4, use the definition of consistency to show that Yn, the nth order statistic, is a consistent estimator of the parameter β. Example 10.4 If X1, X2, . . . , Xn constitute a random sample from a uniform population with α = 0, show that the
Substituting “asymptotically unbiased” for “ unbiased” in Theorem 10.3, show that X + 1 / n+ 2 is a consistent estimator of the binomial parameter θ.
Use the result of Example 8.4 on page 253 to show that for random samples of size n = 3 the median is a biased estimator of the parameter θ of an exponential population.
Substituting “asymptotically unbiased” for “ unbiased” in Theorem 10.3, use this theorem to rework Exercise 10.35.
To show that an estimator can be consistent with-out being unbiased or even asymptotically unbiased, consider the following estimation procedure: To estimate the mean of a population with the finite variance σ2, we first take a random sample of size n. Then we randomly draw one of n slips of paper
If X1, X2, . . . , Xn constitute a random sample of size n from an exponential population, show that is a sufficient estimator of the parameter θ.
If X1 and X2 are independent random variables having binomial distributions with the parameters θ and n1 and θ and n2, show that X1 + X2 / n1 + n2 is a sufficient estimator of θ.
In reference to Exercise 10.43, is X1 + 2X2 / n1 + 2n2 a sufficient estimator of θ?
After referring to Example 10.4, is the nth order statistic, Yn, a sufficient estimator of the parameter β?
If X1 and X2 constitute a random sample of size n = 2 from a Poisson population, show that the mean of the sample is a sufficient estimator of the parameter λ.
If X1, X2, and X3 constitute a random sample of size n = 3 from a Bernoulli population, show that Y = X1 + 2X2 + X3 is not a sufficient estimator of θ. Consider special values of X1, X2, and X3.)
If X1, X2, . . . , Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + · · · + Xn is a sufficient estimator of the parameter θ.
Show that the estimator of Exercise 10.5 is a sufficient estimator of the variance of a normal population with the known mean µ.Exercise 10.5Show that X̅ is a minimum variance unbiased estimator of the mean µ of a normal population.
Given a random sample of size n from a population that has the known mean µ and the finite variance σ2, show that
Given a random sample of size n from a beta population with β = 1, use the method of moments to find a formula for estimating the parameter α.
If X1, X2, . . . , Xn constitute a random sample of size n from a population given byFind estimators for ∂ and θ by the method of moments. This distribution is sometimes referred to as the two-parameter exponential distribution, and for θ = 1 it is the distribution of Example 10.3.
Given a random sample of size n from a continuous uniform population, use the method of moments to find formulas for estimating the parameters α and β.
Consider N independent random variables having identical binomial distributions with the parameters θ and n = 3. If no of them take on the value 0, n1 take on the value 1, n2 take on the value 2, and n3 take on the value 3, use the method of moments to find a formula for estimating θ.
Use the method of maximum likelihood to rework Exercise 10.53. In exercise Given a random sample of size n from a Poisson population, use the method of moments to obtain an estimator for the parameter λ.
Use the results of Theorem 8.1 on page 233 to show that X̅2 is an asymptotically unbiased estimator of µ2.
Use the method of maximum likelihood to rework Exercise 10.54. In exercise Given a random sample of size n from a beta population with β = 1, use the method of moments to find a formula for estimating the parameter α.
X1, X2, . . . , Xn constitute a random sample of size n from a gamma population with α = 2, use the method of maximum likelihood to find a formula for estimating β.
Given a random sample of size n from a normal population with the known mean µ, find the maximum likelihood estimator for σ.
If X1, X2, . . . , Xn constitute a random sample of size n from a geometric population, find formulas for estimating its parameter α by using(a) The method of moments;(b) The method of maximum likelihood.
Given a random sample of size n from a Rayleigh population (see Exercise 6.20 on page 184), find an estimator for its parameter α by the method of maximum likelihood.
Given a random sample of size n from a Pareto population (see Exercise 6.21 on page 184), use the method of maximum likelihood to find a formula for estimating its parameter α.
Use the method of maximum likelihood to rework Exercise 10.56.In exerciseIf X1, X2, . . . , Xn constitute a random sample of size n from a population given byFind estimators for ∂ and θ by the method of moments. This distribution is sometimes referred to as the two-parameter exponential
Use the method of maximum likelihood to rework Exercise 10.57.
Use the method of maximum likelihood to rework Exercise 10.58. In exercise Given a random sample of size n from a continuous uniform population, use the method of moments to find formulas for estimating the parameters α and β.
Given a random sample of size n from a gamma population with the known parameter α, find the maximum likelihood estimator for (a) β; (b) t = (2β – 1) 2.
Show that X + 1 / n + 2 is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased?
If V1, V2, . . . , Vn and W1, W2, . . . , Wn are independent random samples of size n from normal populations with the means µ1 = α + β and µ2 = α – β and the common variance σ2 = 1, find maximum likelihood estimators for α and β.
If V1, V2, . . . , Vn1 and W1, W2, . . . , Wn2 are independent random samples of sizes n1 and n2 from normal populations with the means µ1 and µ2 and the common variance σ2, find maximum likelihood estimators for µ1, µ2, and σ2.
Let X1, X2, . . . , Xn be a random sample of size n from the uniform population given byShow that if Y1 and Yn are the first and nth order statistic, any estimator Θ such thatCan serve as a maximum likelihood estimator of θ. This shows that maximum likelihood estimators need not be unique.
With reference to Exercise 10.72, check whether the following estimators are maximum likelihood estimators of θ:(a) 1/2 (Y1 + Yn); (b) 1/3 (Y1 + 2Y2).In exerciseLet X1, X2, . . . , Xn be a random sample of size n from the uniform population given byShow that if Y1 and Yn are the first and nth
Making use of the results of Exercise 6.29 on page 184, show that the mean of the posterior distribution of Θ given on page 304 can be written asThat is, as a weighted mean of x/n and θ0, where θ0 and σ20 are the mean and the variance of the prior beta distribution of Θ and
In Example 10.19 the prior distribution of the parameter of the binomial distribution was a beta distribution with α = β = 40. Use Theorem 6.5 on page 182 to find the mean and the variance of this prior distribution and describe its shape.
Show that the mean of the posterior distribution of M given in Theorem 10.6 can be written asThat is, as a weighted mean of x and µ0, where
If X has a Poisson distribution and the prior distribution of its parameter Λ(capital Greek lambda) is a gamma distribution with the parameters α and β, show that (a) The posterior distribution of given X = x is a gamma distribution with the parameters α + x and β/β + 1 ; (b) The
With reference to Example 10.3, find an unbiased estimator of d based on the smallest sample value (that is, on the first order statistic, Y1).Example 10.3If X1, X2, . . . , Xn constitute a random sample from the population given byShow that X̅ is a biased estimator of ∂.
On 12 days selected at random, a city’s consumption of electricity was 6.4, 4.5, 10.8, 7.2, 6.8, 4.9, 3.5, 16.3, 4.8, 7.0, 8.8, and 5.4 million kilowatt- hours. Assuming that these data may be looked upon as a random sample from a gamma population, use the estimators obtained in Example 10.14 to
Certain radial tires had useful lives of 35,200, 41,000, 44,700, 38,600, and 41,500 miles. Assuming that these data can be looked upon as a random sample from an exponential population, use the estimator obtained in Exercise 10.51 to estimate the parameter θ.
The size of an animal population is sometimes estimated by the capture-recapture method. In this method, n1 of the animals are captured in the area under consideration, tagged, and released. Later, n2 of the animals are captured, X of them are found to be tagged, and this information is used to
Among six measurements of the boiling point of a silicon compound, the size of the error was 0.07, 0.03, 0.14, 0.04, 0.08, and 0.03○C. Assuming that these data can be looked upon as a random sample from the population of Exercise 10.55, use the estimator obtained there by the method of moments to
Not counting the ones that failed immediately, certain light bulbs had useful lives of 415, 433, 489, 531, 466, 410, 479, 403, 562, 422, 475, and 439 hours. Assuming that these data can be looked upon as a random sample from a two-parameter exponential population, use the estimators obtained in
Data collected over a number of years show that when a broker called a random sample of eight of her clients, she got a busy signal 6.5, 10.6, 8.1, 4.1, 9.3, 11.5, 7.3, and 5.7 percent of the time. Assuming that these figures can be looked upon as a random sample from a continuous uniform
In a random sample of the teachers in a large school district, their annual salaries were $ 23,900, $ 21,500, $ 26,400, $ 24,800, $ 33,600, $ 24,500, $ 29,200, $ 36,200, $ 22,400, $ 21,500, $ 28,300, $ 26,800, $ 31,400, $ 22,700, and $ 23,100. Assuming that these data can be looked upon as a random
With reference to Example 10.4, find an unbiased estimator of β based on the smallest sample value (that is, on the first order statistic, Y1). Example 10.4 If X1, X2, . . . , Xn constitute a random sample from a uniform population with α = 0, show that the largest sample value (that is, the nth
Every time Mr. Jones goes to the race track he bets on three races. In a random sample of 20 of his visits to the race track, he lost all his bets 11 times, won once 7 times, and won twice on 2 occasions. If θ is the probability that he will win any one of his bets, estimate it by using the
On 20 very cold days, a farmer got her tractor started on the first, third, fifth, first, second, first, third, seventh, second, fourth, fourth, eighth, first, third, sixth, fifth, second, first, sixth, and second try. Assuming that these data can be looked upon as a random sample from a geometric
The I.Q.’s of 10 teenagers belonging to one ethnic group are 98, 114, 105, 101, 123, 117, 106, 92, 110, and 108, whereas those of 6 teenagers belonging to another ethnic group are 122, 105, 99, 126, 114, and 108. Assuming that these data can be looked upon as independent random samples from
The output of a certain integrated- circuit production line is checked daily by inspecting a sample of 100 units. Over a long period of time, the process has maintained a yield of 80 percent, that is, a proportion defective of 20 percent, and the variation of the proportion defective from day to
Records of a university (collected over many years) show that on the average 74 percent of all incoming freshmen have I.Q.’s of at least 115. Of course, the percentage varies somewhat from year to year, and this variation is measured by a standard deviation of 3 percent. If a sample check of 30
With reference to Example 10.20, find P (712 < M < 725| x̅ = 692).Example 10.20A distributor of soft- drink vending machines feels that in a supermarket one of his machines will sell on the average µ0 = 738 drinks per week. Of course, the mean will vary somewhat from market to market, and
A history professor is making up a final examination that is to be given to a very large group of students. His feelings about the average grade that they should get is expressed subjectively by a normal distribution with the mean µ0 = 65.2 and the standard deviation σ0 = 1.5. (a) What prior
An office manager feels that for a certain kind of business the daily number of incoming telephone calls is a random variable having a Poisson distribution, whose parameter has a prior gamma distribution with α = 50 and β = 2. Being told that one such business had 112 incoming calls on a given
How large a random sample is required from a population whose standard deviation is 4.2 so that the sample estimate of the mean will have an error of at most 0.5 with a probability of 0.99?
If x is a value of a random variable having an exponential distribution, find k so that the interval from 0 to kx is a (1 – α) 100% confidence interval for the parameter θ.
Verify the result on page 318, which expresses T in terms of X1, X2, and Sp.
By solvingFor Î¸, Show that Are (1 €“ Î±) 100% confidence limits for Î¸.
Use the result of Exercise 11.15 to show that when n1 = n2 = n, we can be at least (1 €“ a) 100% confident that the error that we make when using Î¸1 €“ Î¸2 as an estimate of Î¸1 €“ Î¸2 is less than e when
Fill in the details that led from the probability on page 325 to the confidence-interval formula of Theorem 11.10.
For large n, the sampling distribution of S is some-times approximated with a normal distribution having the mean Ïƒ and the variance Ïƒ2/2n (see Exercise 8.28 on page 250). Show that this approximation leads to the following (1 €“ Ïƒ) 100% large- sample
If x1 and x2 are the values of a random sample of size 2 from a population having a uniform density with Î± = 0 and Î² = Î¸, find k so thatIs a (1 €“ Î±) 100% confidence interval for Î¸ when (a) Î± ‰¤
A district official intends to use the mean of a random sample of 150 sixth graders from a very large school district to estimate the mean score that all the sixth graders in the district would get if they took a certain arithmetic achievement test. If, based on experience, the official knows that
A medical research worker intends to use the mean of a random sample of size n = 120 to estimate the mean blood pressure of women in their fifties. If, based on experience, he knows that σ = 10.5 mm of mercury, what can he assert with probability 0.99 about the maxi-mum error?
With reference to Exercise 11.22, suppose that the research worker takes his sample and gets = 141.8 mm of mercury. Construct a 98% confidence interval for the mean blood pressure of women in their fifties.
A study of the annual growth of certain cacti showed that 64 of them, selected at random in a desert region, grew on the average 52.80 mm with a standard deviation of 4.5 mm. Construct a 99% confidence interval for the true average annual growth of the given kind of cactus.
If a sample constitutes an appreciable portion of a population, that is, more than 5 percent of the population according to the rule of thumb given on page 239, the formulas of Theorems 11.1 and 11.2 must be modified by using the variance formula of Theorem 8.6 on page 238 instead of that of
Use the modification suggested in Exercise 11.26 to rework Exercise 11.21, given that there are 900 sixth graders in the school district.
An efficiency expert wants to determine the aver-age amount of time it takes a pit crew to change a set of four tires on a race car. Use the formula for n in Exercise 11.6 to determine the sample size that is needed so that the efficiency expert can assert with probability 0.95 that the sample mean
In a study of television viewing habits, it is desired to estimate the average number of hours that teenagers spend watching per week. If it is reasonable to assume that σ = 3.2 hours, how large a sample is needed so that it will be possible to assert with 95% confidence that the sample mean is
Making use of the methods of Section 8.7, it can be shown that for a random sample of size n = 2 from the population of Exercise 11.2, the distribution of the sample range is given byUse this result to find c so that R
The length of the skulls of 10 fossil skeletons of an extinct species of bird has a mean of 5.68 cm and a standard deviation of 0.29 cm. Assuming that such measurements are normally distributed, find a 95% confidence interval for the mean length of the skulls of this species of bird.
A major truck stop has kept extensive records on various transactions with its customers. If a random sample of 18 of these records shows average sales of 63.84 gallons of diesel fuel with a standard deviation of 2.75 gallons, construct a 99% confidence interval for the mean of the population
Independent random samples of sizes n1 = 16 and n2 = 25 from normal populations with σ1 = 4.8 and σ2 = 3.5 have the means 1 = 18.2 and 2 = 23.4. Find a 90% confidence interval for µ1 – µ2.
A study of two kinds of photocopying equipment shows that 61 failures of the first kind of equipment took on the average 80.7 minutes to repair with a standard deviation of 19.4 minutes, whereas 61 failures of the second kind of equipment took on the average 88.1 minutes to repair with a standard
Twelve randomly selected mature citrus trees of one variety have a mean height of 13.8 feet with a standard deviation of 1.2 feet, and 15 randomly selected mature citrus trees of another variety have a mean height of 12.9 feet with a standard deviation of 1.5 feet. Assuming that the random samples
The following are the heat- producing capacities of coal from two mines (in millions of calories per ton):Assuming that the data constitute independent random samples from normal populations with equal variances, construct a 99% confidence interval for the difference between the true average
To study the effect of alloying on the resistance of electric wires, an engineer plans to measure the resistance of n1 = 35 standard wires and n2 = 45 alloyed wires. If it can be assumed that σ1 = 0.004 ohm and σ2 = 0.005 ohm for such data, what can she assert with 98% confidence about the
A sample survey at a supermarket showed that 204 of 300 shoppers regularly use coupons. Use the large-sample confidence-interval formula of Theorem 11.6 to construct a 95% confidence interval for the corresponding true proportion.
Show that the (1 €“ Î±) 100% confidence intervalIs shorter than the (1 €“ Î±) 100% confidence interval

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