New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
business
nonparametric statistical inference
Probability And Statistical Inference 3rd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions
A multiple-choice exam gives five answers to each of its n questions, only one being correct. Assume that a student who does not know the answer chooses randomly and is correct with probability 0.2. Let θ be the number of questions to which the student knows the answers, and let X be the number of
An urn contains six balls, r red and 6 − r blue. Two balls are chosen without replacement. Find the most powerful test of H0 : r = 3 against the alternative H1 : r = 5, with a size as close to α = 0.05 as possible. Find the probability of a type II error for all r= 3.
The sample space of a test statistic X has five values: a,b,c,d,e. Test the H0 : f = f0 against Ha : f = f1, where distributions f0 and f1 are given by the table Xa b c d e f0 0.2 0.2 0.0 0.1 0.5 f1 0.2 0.4 0.3 0.0 0.1
Let X1, ...,Xn be a random sample from EXP(λ) distribution. Null hypothesis H0 : λ = λ0 is tested against the alternative H1 : λ = λ1, where λ1 > λ0. Compare the power functions of the two tests: (i) the most powerful test and (ii) the most powerful test based on the statistic X1:n. Assume
Assume that X has a N(2,σ2) distribution. Find the best critical region for testing H0 : σ2 = 2 against: (i) H1 : σ2 = 4. (ii) H1 : σ2 = 1.
Let X have a negative binomial distribution with parameters r and p. Find the most powerful test of H0 : r = 2,p = 1/2 against H1 : r = 4,p = 1/2 at significance level α = 0.05. Find probability of type II error. Use randomized test if necessary.
A single observation X is taken from a BETA(a,b) distribution. Find the most powerful test of the null hypothesis H0: a = b = 1, against the alternative H1:(i) a = b = 5. (ii) a = 2,b = 3 (iii) a = b = 1/2. Use significance level α = 0.05.
Let X1, ...,X10 be a random sample from a POI(θ) distribution. (i) Find the best critical region for testing H0 : θ = 0.1 against H1 : θ = 0.5 at the significance levelα = 0.05. (ii) Determine the size of the test in (i).
A coin is thrown independently 40 times to test that the probability of heads θ is 0.5 against the alternative that it is greater than 0.5. The test rejects the null hypothesis if more than 24 heads are observed in these 40 tosses. (i) Use the CLT with continuity correction to approximate the
Let X1, ...,X9 be a random sample from the N(μ, 1) distribution. To test the hypothesis H0 : μ ≤ 0 against H1 : μ > 0, one uses the test “reject H0 if 3 ≤ X ≤ 5.” Find the power function and show that this is a bad test.
A large box is full of chips in three colors: red, black, and green. The hypothesis to be tested is that chips in the box are mixed in equal proportions with the alternative that they are not. Four chips are to be selected with replacement and the null hypothesis will be rejected if at least one
An urn contains five balls, r red and 5 − r white. The null hypothesis states that all balls are of the same color (i.e., H0 : r = 0 or r = 5). Suppose that we take a sample of size 2 and reject H0 if the balls are of different colors. Find the power of this test for r = 0, ..., 5 if the sample
Let X1,X2 be a random sample of size 2 from the U[0,θ] distribution. We want to test H0 : θ = 3 against H1 : θ = 2 (observe that H0 and H1 do not exhaust all possibilities). (i) H0 will be rejected if X
Consider three tests C1,C2, and C3 of the same hypothesis, performed independently(e.g., for each of these tests the decision is based on a different sample). Consider now the following three procedures:A: Reject H0 only if all three tests reject it; otherwise, accept H0, B: Reject H0 only if at
Consider the following procedure for testing the hypothesis H0 : p ≥ 0.5 against the alternative H1 : p < 0.5 in BIN(10,p) distribution. We take observation X1, and reject H0 if X1 = 0 or accept H0 if X1 ≥ 9; otherwise, we take another observation X2 (with the same distribution as X1 and
Let X1,X2 be a random sample from the U[θ,θ + 1] distribution. In the test of H0 : θ = 0 against H1 : θ > 0, H0 is rejected when X1 + X2 > k. Find the power function of the test that has probability of the type I error equal 0.05.
to obtain 90% bootstrap intervals (t and percentile)for the mean width-to-length ratio of Shoshoni handicraft. Use 2,000 bootstrap samples. As statistic use X and the 10% trimmed mean. Compare your results.
Use the data in Problem
(i) Given a random sample X1, ...,Xn obtain a 90% CI for the mean μ and then obtain a 90% bootstrap t interval and a 90% percentile interval for the same parameter μ. In addition, obtain two bootstrap intervals (t and percentile) for μ using a 10% trimmed mean as a statistic. As your sample use
A sample of 200 trees in a forest has been inspected for a presence of some bugs, out of which 37 trees were found to be infested. (i) Assuming a binomial model, give a 90% confidence interval for the probability p of a tree being infested. Use the exact and the approximate formulas. (ii) Usually,
Suppose that the arrivals at a checkout counter in a supermarket (i.e., times of arriving at the counter or joining the queue, whichever is earlier) form a Poisson process with arrival rate λ. Counting from noon, the thirteenth customer arrived at 12:18 p.m. Find a 90% CI for: (i) λ. (ii) The
Let X1, ...,Xn be a random sample from EXP(λ) distribution. (i) Find an exact 100(1−α)% confidence interval for θ = 1/λ. (ii) Assuming that you have m such intervals, each based on a random sample of the same size n, find the probability that at least k of them (k
Let X1, ...,Xn be a random sample from some continuous distribution. If the confidence interval for the median M is (X1:n,Xn:n) find the smallest n such that the confidence level 1 − α is at least 0.95.
Based on a random sample X1, ...,Xn from the U[0,θ] distribution find: (i)P{Xn:n
Suppose that the lifetime T of a certain kind of device (e.g., a fuel pump in a car)has an EXP(λ) distribution. The observed lifetimes of a sample of the devices are 350,727, 615,155, 962 (in days). Find a 95% CI’s for: (i) λ. (ii) E(T). (iii) The standard deviation of the lifetime of the
Based on a random sample of size n selected from the WEI(4,θ) distribution, derive a 95% confidence interval for θ based on X1:n.
(i) Use the large sample distribution of MLE of mean λ in Poisson distribution to construct an approximate (1 − α)-level CI for λ. (ii) Assuming that the numbers of new cars of a given make sold per week in 15 consecutive weeks—5, 5, 6, 3, 5, 8, 1, 4, 7, 7, 5, 4, 3, 0, 9—form a random
Suppose that the largest observation recorded in a sample of size n = 35 from a distribution uniform on [0,θ] is 5.17. Find a 90% CI for θ.
Find the probability that the length of a 95% confidence interval for the mean of normal distribution with unknown σ is less than σ, n = 25.
Obtain a (1 − α)100% CI for θ if X1, ...,Xn is a random sample from a: (i) N(μ,θ)distribution with μ and θ unknown. (ii) BETA(1,θ) distribution.
Let X1, ...,Xn be a random sample from N(μ,σ2) distribution with both parameters unknown. Let Lα be the length of the shortest confidence interval for μ on confidence level 1 − α. (i) Find E(L2α) as a function of n,σ2 and α. (ii) Find the smallest n such that E(L2α) ≤ σ2/2 for a given
A large company wants to estimate the fraction p of its employees who participate in a certain health program. It has been decided that if p is below 25%, a special promotion campaign will be launched. In a random sample of 85 employees the number of those who participated in the program was 16.
Seven measurements of the concentration of some chemical in cans of tomato juice are 1.12, 1.18, 1.08, 1.13, 1.14, 1.10, 1.07. Assume that these numbers represent a random sample from the distribution N(θ,σ2). (i) Find the shortest 95% and 99%CI’s for θ, if σ2 is unknown. (ii) Answer part (i)
Based on a random sample 1.23, 0.36, 2.13, 0.91, 0.16, 0.12 selected from the GAM(2.5,θ) distribution, find an exact 95% CI for parameter θ.
Let X1, ...,Xn be a random sample selected from the Pareto distribution with density f(x; θ) = θ2θx−(θ+1) for x ≥ 2. Find: (i) The sufficient statistic for θ. (ii) An exact 95% CI for θ.
Let X1, ...,X16 and Y1, ...,Y12 be random samples from N(θ, 15) and N(θ, 20)distributions, respectively. Find the 95% CI for θ if x = 20 and y = 18.
Let X1, ...,Xn be a random sample from the distribution with density f(x; θ) =θ(x + 1)−θ−1 for x > 0,θ> 1. Find the exact 90% confidence interval for θ.
Let X1, ...,Xn be a random sample from the U(0,θ) distribution. Show that statistic T = Xn:n/X1:n is ancillary for θ.
Let X1,X2 be a random sample of size n = 2 from the EXP(λ) distribution. Show that statistic T = X1/X2 is ancillary for λ.
Let X1, ...,Xn be a random sample from the U(θ,θ + 1) distribution. Show that statistic T = Xn:n − X1:n is ancillary for θ.
Let X1,X2 be a random sample of size 2 from the POI(λ) distribution. Show that statistic T = X1 + 2X2 is not sufficient for λ.
Let X1, ...,Xn be a random sample from the EXP(λ) distribution. Suppose that only first k order statistics X1:n, ...,Xk:n are observed. Find a minimal sufficient statistic for λ.
Suppose that a random sample is taken from a distribution with density f(x; θ) =2x/θ2 for 0 ≤ x ≤ θ and f(x; θ)=0 otherwise. Find the MLE of the median of this distribution, and show that this estimator is a minimal sufficient statistic.
Show that the family of GAM(α,λ) distributions is in the exponential class, and find the minimal jointly sufficient statistics.
Check if the following families of distributions are in the exponential class: (i)POI(λ). (ii) EXP(λ). (iii) NBIN(r,θ), r known. (iv) BETA(θ1,θ2). (v) WEI(k,θ), k known. (vi) WEI(θ,/), / known.
Let X1, ...,Xn be a random sample from the distribution with a density f(x; λ,θ) = λe−λ(x−θ) for x ≥ θ and 0 otherwise. Determine a pair of jointly sufficient statistics for parameters λ and θ.
Show that the N(0,θ) family is not complete.
Find a sufficient statistic for θ if observations are uniformly distributed on the set of integers 0, 1, . . . , θ.
Generalizing Example 11.39, let X1, ...,Xn be n independent Bernoulli trials.Show that T = n i=1 Xi is sufficient for probability of success p by finding the joint distribution of (X1, ...,Xn) given T = t. Find the marginal distribution P{X1 = 1|T = t}.
Find sufficient statistics (or show that they do not exist) for parameter θ in the following distributions: (i) f(x,θ)=(x/θ2)e−x2/2θ2 for x > 0 (Rayleigh). (ii) f(x,θ) =(1/2θ)e−|x|θ (double exponential). (iii) BETA(θ, 2θ). (iv) U[θ, 2θ]. (v) GAM(θ,θ).(vi) f(x,θ) = {π[1 + (x −
Let X1, ...,Xn be a random sample from EXP(θ) distribution. Find the MLE of the second factorial moment E(X(X − 1)) and determine its asymptotic distribution.
Let X1, ...,Xn be independent observations from a distribution with density f(x,θ)=2x/θ2, for 0 ≤ x ≤ θ and 0 otherwise. Find the MME of θ and approximate its distribution.
Let X1, ...,Xn be a random sample from a distribution with density f(x,θ) =θ(1 + x)−(1+θ) and for x > 0 and 0 otherwise, θ ≥ 1. Find the MME of θ.
Let X1, ...,Xn be a random sample from U[θ,θ + 1] distribution. (i) Show that T = c(Xn:n − 1) + (1 − c)X1:n, 0
For independent variables Y1, ...,Yn with distribution N(α + βxi,σ2), where x1, ...,xn are fixed, show that the LS-estimator and ML-estimator of θ = (α,β)coincide.
Y1, ...,Yn are independent variables. Assuming that x1, ...,xn are such that(xi − x)2 > 0, compare the MSE’s of the MLE and LS-estimators of parameterθ > 0, if: (i) Yi ∼ EXP(xi/θ). (ii) Yi ∼ POI(θxi).
Let X1, ...,Xm be a random sample from the N(μ1,σ2 1) distribution and let Y1, ...,Yn be a random sample from the N(μ2,σ2 2) distribution, with Xi’s being independent from Yj ’s. Find the MLE of: (i) μ1,μ2,σ2 if σ1 = σ2 = σ. (ii) μ,σ2 1,σ2 2where μ1 = μ2 = μ.
Let X1, ...,Xn be a random sample from a log-normal distribution with parameters μ and σ2 [this means that log Xi ∼N(μ,σ2)]. Find the MLE of μ and σ2.
Suppose that the median of 20 observations, taken from a normal distribution with an unknown mean and variance, is 5 and that only one observation differs from the median by more than 3. Suggest an estimate of the probability that the next two observations will both be between 4 and 5.
For n observations taken from the U[0,θ] distribution, let Un be the number of the ones that are less than 3. Find the MLE of θ.
Find the MLE of the mean of a U[θ1,θ2] distribution based on a random sample of size n.
Let X1,X2 be a random sample from a N(μ,σ2) distribution with μ and σ2 unknown. Find the MLE of σ2 if the only available information is that the difference between observations equals 3.
Let X1, ...,Xn be a random sample from N(μ,σ2) distribution, μ is known. Find the MLE of σ: (i) Directly. (ii) First finding the MLE of variance σ2 and then using the invariance property.
Find the MME and MLE of the standard deviation of a Poisson distribution.
Let X1, ...,Xn be a random sample from POI(λ) distribution. Find the MLE of λassuming that: (i) X1 + ··· + Xn > 0. (ii) X1 + ··· + Xn = 0.
Two independent Bernoulli trials resulted in one failure and one success. What is the MLE of the probability of success θ if it is known that: (i) θ is at most 1/4. (ii) θexceeds 1/4.
Suppose that there were 15 successes in 24 Bernoulli trials. Find the MLE of the probability of success θ if it is known that θ ≤ 1/2.
Let X1, ...,Xn be a random sample from a distribution with density f(x,θ) =3θ−3x2 for 0 ≤ x ≤ θ. (i) Find the MLE of θ. (ii) Find the MME of θ. (iii) Approximate the distribution of T = 3X. (iv) Obtain the MSE of Ta = aX as an estimator of θ.
Let X1, ...,Xn be a random sample from N(θ,σ2) distribution, θ = 0,σ2 is known. Determine the asymptotic distribution of (X)1/2 using the invariance property of MLEs.
Find the distribution of the MLE of the probability of success θ based on two Bernoulli trials.
A single observation of a random variable X with a geometric distribution results in X = k. Find the MLE of the probability of success θ if: (i) X is the number of failures preceding the first success. (ii) X is the number of trials up to and including the first success.
Some phenomena (e.g., headway in traffic) are modeled to have a distribution of a sum of a constant and an exponential random variable. Then the density of X has the form f(x; a,b) = 0 for x 0 and b > 0 are two parameters. Find: (i) The MME of θ = (a,b). (ii) The MLE of θ.
(Bragging Tennis Player) As in Example 11.34, consider tennis players A and B who from time to time play matches against each other. The probability that A wins a set against B is p.Suppose now that we do not have complete data on all matches between A and B;we learn only of A’s victories, so we
Let X1, ...,Xn be a random sample from Poisson distribution with mean λ. Find the MLE of P(X = 0).
Let X1, ...,Xn be a random sample from the distribution uniform on the union of the two intervals: [−2, −1] and [0, θ]. Find: (i) The MME of θ. (ii) The MLE of θ.(iii) The MLE of θ if positive Xi’s are recorded exactly, and negative Xi’s can only be counted. (iv) The MLE of θ if Xi’s
Find the MME of parameter θ in the distribution with density f(x,θ) =(θ + 1)x−(θ+2), for x > 1,θ> 0.
Let X1, ...,Xn be a random sample from GAM(α,λ) distribution. Find: (i) The MME of θ = (α,λ), using the first two moments. (ii) The MME of α when λ is known, and the MME of λ when α is known.
Let X1, ...,Xn be a random sample form the POI(λ) distribution. Find the Cramér–Rao lower bound for variances of unbiased estimators of P(X ≤ 1) = (1 + λ)e−λ.
Show that the estimator T = X satisfies relation (11.46) and determine functions γ1 and γ2 if a random sample X1,X2, ...,Xn is selected from: (i) N(θ,σ2) distribution with σ known. (ii) BIN(1,θ) distribution.
Let X1, ...,Xn be a random sample from a Bernoulli distribution with an unknown p. Show that the variance of any unbiased estimator of (1 − p)2 must be at least 4p(1 − p)3/n.
Let X1,X2 be a random sample of size 2 from N(μ,σ2) distribution. Determine the amount of information about μ and about σ2 contained in: (i) X1 + X2. (ii) X1 − X2.
Find, if it exists, the Fisher information I(θ) in a random sample of size n from:(i) The Cauchy distribution with density f(x,θ) = {π[1 + (x − θ)2]}−1, −∞
Let X1, ...,Xn be a random sample from EXP(λ) distribution. Propose an efficient estimator of 1/λ and determine its variance
Let X have EXP(λ) distribution. Find the Fisher information I(λ).
Let X2k be the sample mean of 2k independent observations from a normal distribution with mean θ and known variance σ2. Find the efficiency of Xk (i.e., of estimator that uses only half of the sample).
Let X1, ...,Xn be a random sample from the distribution with density f(x,θ) =2x/θ2 for 0 ≤ x ≤ θ and 0 otherwise. Determine the MSE of ˆθ = Xn:n.
Let X1, ...,X4 be a random sample from U[0,θ] distribution. Compare the mean squared errors of four estimators of θ: T1 = 5X1:4, T2 = (5/2)X2:4, T3 = (5/3)X3:4, and T4 = (5/4)X4:4.
Let U = X1:n and let V = Xn:n in a random sample from U[θ − 1,θ + 1] distribution. (i) Show that X and (U + V )/2 are both unbiased estimators of θ. (ii) Determine the MSE’s of estimators in (i).
Let X1, ...,Xn be a random sample from a N(μ,σ2) distribution (μ and σ2 are unknown), and let S2 = 1 nn i=1(Xi − X)2, S2 1 = 1 n − 1n i=1(Xi − X)2 be two estimators of σ2. (i) Compare the MSE’s of S2 and S2 1 . (ii) Consider S2 k = kn i=1(Xi − X)2 as estimators of σ2 and find k
Let Y1, ...,Yn be a random sample of size n from the discrete distribution with probability function f(y; θ) = θ(1 − θ)y−1,y = 1, ... . Compare the MSE’s of two estimators of τ = 1/θ: T1 = Y and T2 = [n/(n + 1)]Y .
Let X1, ...,Xn be n Bernoulli trials with probability of success θ, and let S = n i=1 Xi. Compare the mean squared errors of two estimators of θ: T1 = S/n and T2 = (S + 1)/(n + 2).
Let X1, ...,Xn be a random sample from EXP(1/θ) distribution. Obtain the mean squared errors of two estimators of θ: T1 = X and T2 = [n/(n + 1)]X.
Let X1, ...,Xn be a random sample from a N(θ,σ2) distribution with σ2 known.Show that the estimator T of θ, defined as T(X1, ...,Xn) ≡ 3 (T = 3 regardless of the observations), is admissible.
Let T1 and T2 be two unbiased estimators of θ with variances σ2 1,σ2 2, respectively.Find values a and b such that: (i) Estimator aT1 + bT2 is unbiased. (ii) Unbiased estimator aT1 + bT2 has a minimum variance assuming that T1 and T2 are independent.(iii) Unbiased estimator aT1 + bT2 has a
Let Xn be a relative frequency of success in n Bernoulli trials. Use the Delta method to find the limiting distribution of: (i) g(Xn) = Xn(1 − Xn). (ii)g(Xn) = Xn(1 − Xn). (iii) g(Xn) = (1 − Xn)/Xn.
Let Xn be a sample mean in a random sample X1, ...,Xn from POI(λ) distribution and let Tn = n i=1 Xi. Use the Delta method to find the limiting distribution of:(i) g(Xn) = √n(Xn − λ). (ii) g(Xn)=1/Xn. (iii) g(Tn) = log(Tn). (iv) g(Tn ) =Tn. [Hint: Tn/n has approximately N(λ,λ/n)
A die is unbalanced in such a way that the probability of tossing k (k = 1, ..., 6) is proportional to k. You pay $4 for a toss, and win $k if you toss k. Find the approximate probability that you are ahead after 100 tosses.
Referring to Example 7.39, assume that a man’s shoe has an average length of1 foot and σ = 0.1 foot. Find the (approximate) probability that the mean of 16 lengths of men’s shoes exceed1 foot by more than1 inch.
Let X1, ...,Xn be a random sample from EXP(λ) distribution. Show that there exist normalizing constants An and Bn such that (X1 + ··· + Xn − An)/Bn has asymptotically N(0, 1) distribution.
A rate at which patients arrive to an urgent care center in the evening is one per 6 minutes. Assuming that the number of arrivals follows Poisson distribution, find the probability that more than 35 patients but no more than 50 will arrive to the center on a given day between 6 p.m. and 10 p.m.
Showing 700 - 800
of 5397
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Last
Step by Step Answers
Get 50% OFF
Claim Now
