Math 562 Homework 1 August 29, 2006 Dr. Ron Sahoo

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1 Math 56 Homework August 9, 006 Dr. Ron Sahoo He who labors diligently need never despair; for all things are accomplished by diligence and labor. Menander of Athens Direction: This homework worths 60 points and is due on September 4, 006. In order to receive full credit, answer each problem completely and must show all work.. Seven observations are drawn from a population with an unknown continuous distribution. What is the probability that the least and the greatest observations bracket the median?. If the random variable X has the density ( x) for 0 x what is the probability that the larger of independent observations of X will exceed? 3. Let X, X, X 3, X 4, X 5 be a random sample from the uniform distribution on the interval (0, ), where is unknown, and let X max denote the largest observation. For what value of the constant k, the expected value of the random variable kx max is equal to? 4. Five observations have been drawn independently and at random from a continuous distribution. What is the probability that the next observation will be less than all of the first 5? 5. Let X and Y be two independent random variables with identical probability density given by 3 x for 0 x 3 0 elsewhere, for some > 0. What is the probability density of W = min{x, Y }?

2 6. Let X, X, X 3 be a random sample of size 3 from a standard normal distribution. X X X 3. X +X +X 3 Find the sampling distribution of the statistics X +X +X 3 X +X +X 3 7. Suppose X, X,..., X n is a random sample from a normal distribution with mean µ and variance σ. If X = n n i= X i and Σ = n n i= ( Xi X ), and X n+ is an additional observation, what is the value of m so that the statistics m( X X n+ ) Σ has a t-distribution. 8. Let X, X, X 3, X 4 be a random sample of size 4 from a standard normal population. Find the distribution of the statistic X +X 4 X +X3 9. Let X, X,..., X n be a random sample from a normal distribution with mean µ and variance σ. What is the variance of Σ = n n i= ( Xi X )? 0. A random sample X, X,..., X n of size n is selected from a normal population with mean µ and standard deviation. Later an additional independent observation X n+ is obtained from the same population. What is the distribution of the statistic (X n+ µ) + n i= (X i X), where X denote the sample mean?. Suppose X j = Z j Z j, where j =,,..., n and Z 0, Z,..., Z n are independent and identically distributed with common variance σ. What is the variance of the random variable n n j= X j?. Let X, X,..., X n and Y, Y,..., Y n be two random sample from the independent normal distributions with V ar[x i ] = σ and V ar[y i ] = σ, for i =,,..., n and σ > 0. If U = n i= ( Xi X ) and V = n i= ( Yi Y ), then what is the sampling distribution of the statistic U+V σ? 3. Let X, X,..., X 9 be a random sample of size 9 from a distribution with a probability density x if 0 < x < where 0 < is a parameter. Using the moment method find an estimator for the parameter.. and

3 4. Let X, X,..., X 7 be a random sample of size 7 from a distribution with a probability density x e x if 0 < x < where 0 < is a parameter. Using the moment method find an estimator for the parameter. 5. Let X, X,..., X 8 be a random sample of size 8 from a distribution with a probability density ( + ) x if < x < where 0 < is a parameter. Using the moment method find an estimator for the parameter. Extra problems for graduate students. 6. Let X, X,..., X n be a random sample from a uniform distribution on the interval from 0 to 5. What is the limiting moment generating of X µ σ n as n? 7. Let X, X,..., X n be a random sample of size n from a normal distribution with mean µ and variance. If the 75 th percentile of the statistic W = n i= ( Xi X ) is 8.4, what is the sample size n? 8. Let X, X,..., X n be a random sample of size n from a Bernoulli distribution with probability of success p =. What is the limiting distribution the sample mean X? 9. Let X, X,..., X 995 be a random sample of size 995 from a distribution with probability density e λ λ x x! What is the distribution of 995X? x = 0,,, 3,...,. 0. Let X, X,..., X 9 be a random sample from a uniform distribution on the interval [, ]. Find the probability that the next to smallest is greater than or equal to 4?

4 Math 56 Homework September 4, 006 Dr. Ron Sahoo I hear and I forget. I see and I remember. I do and I understand. Confucius Direction: This homework worths 60 points and is due on October, 006. In order to receive full credit, answer each problem completely and must show all work. Graduate students should do all problems; and the undergraduate students are required to do any 7 problems from the group A and any 8 problems from the group B. GROUP A. Given, the random variable X has a binomial distribution with n = and probability of success. If the prior density of is k if < < h() = what is the Bayes estimate of for a squared error loss if the sample consists of x = and x =.. Suppose two observations were taken of a random variable X which yielded the values and 3. The density for X is if 0 < x < f(x/) = and prior distribution for the parameter is h() = { 3 4 if > 0 otherwise. If the loss is quadratic, then what is the Bayes estimate for? 3. Suppose one observation was taken of a random variable X which yielded the value. The density for X is f(x/µ) = π e (x µ) < x <,

5 and prior distribution of µ is h(µ) = π e µ < µ <. If the loss is quadratic, then what is the Bayes estimate for µ? 4. Let X, X,..., X 5 be a random sample of size 5 from a distribution with probability density if x 3 where > 0. What is the maximum likelihood estimator of? 5. Suppose X and Y are independent random variables each with density { x for 0 < x < 0 otherwise. If k (X + Y ) is an unbiased estimator of, then what is the value of k? 6. Let X, X,..., X 5 be a random sample of size 5 from a distribution with probability density if 0 x where > 0. What is the maximum likelihood estimator of? 7. What is the maximum likelihood estimate of β if the 5 values 4 5, 3,, 3, 5 4 were drawn from the population for which f(x; β) = ( + β)5 ( x ) β? 8. Given, the random variable X has a binomial distribution with n = 3 and probability of success. If the prior density of is k if < < h() = what is the Bayes estimate of for a absolute difference error loss if the sample consists of one obervation x =? 9. Eight independent trials are conducted of a given system with the following results: S, F, S, F, S, S, S, S where S denotes the success and F denotes

6 the failure. What is the maximum likelihood estimate of the probability of successful operation p? 0. Suppose X, X,... are independent random variables, each with probability of success p and probability of failure p, where 0 p. Let N be the number of observation needed to obtain the first success. What is the maximum likelihood estimator of p in term of N? GROUP B. Let T and T be estimators of a population parameter based upon the same random sample. If T i N (, σ i ) i =, and if T = bt + ( b)t, then for what value of b, T is a minimum variance unbiased estimator of?. Let X, X,..., X n be a random sample from a distribution with density x e < x <, where 0 < is a parameter. What is the expected value of the maximum likelihood estimator of? Is this estimator unbiased? 3. A random sample X, X,..., X n of size n is selected from a normal distribution with variance σ. Let S be the unbiased estimator of σ, and T be the maximum likelihood estimator of σ. If 0T 9S = 0, then what is the sample size? 4. Suppose X and Y are independent random variables each with density { x for 0 < x < 0 otherwise. If k (X + Y ) is an unbiased estimator of, then what is the value of k? 5. Let X, X,..., X n be a random sample from a population with probability density if 0 < x < 0 otherwise, where > 0 is an unknown parameter. If X denotes the sample mean, then what should be value of the constant k such that kx is an unbiased estimator of?

7 6. Let X, X,..., X n be a random sample from a population with probability density if 0 < x < 0 otherwise, where > 0 is an unknown parameter. If X med denotes the sample median, then what should be value of the constant k such that kx med is an unbiased estimator of? 7. Let X, X,..., X n be a random sample from a population X with density for 0 x < (+x) + where > 0 is an unknown parameter. What is a sufficient statistic for the parameter? 8. Let X, X,..., X n be a random sample from a population X with density x e x for 0 x < where is an unknown parameter. What is a sufficient statistic for the parameter? 9. Let X, X,..., X n be a random sample from a distribution with density e (x ) for < x < where < < is a parameter. What is the maximum likelihood estimator of? Find a sufficient statistics of the parameter. 0. Let X, X,..., X n be a random sample from a distribution with density e (x ) for < x < where < < is a parameter. Are the estimators X () and X are unbiased estimators of? Which one is more efficient than the other?

8 Math 56 Homework 3 October, 006 Dr. Ron Sahoo Everything should be made as simple as possible, but no simpler. Albert Einstein Direction: This homework worths 60 points and is due on November, 006. In order to receive full credit, answer each problem completely and must show all work. Graduate students should do all problems in groups A and B; and the undergraduate students are required to do any 7 problems from the group A and any 8 problems from the group B. Group C is for extra credit; each problem worths 3 points. GROUP A. Let X, X,..., X n be a random sample from a population with gamma density x β e x Γ(β) f(x;, α) = β for 0 < x < where is an unknown parameter and β > 0 is a known parameter. Show that [ n i= X ] n i χ (nβ), i= X i χ α α (nβ) is a ( α)00% confidence interval for the parameter.. Let X, X,..., X n be a random sample from a population with Weibull density β [ ] x β e xβ for 0 < x < f(x;, α) = where is an unknown parameter and β > 0 is a known parameter. Show that [ n ] n i= Xβ i χ (n), i= Xβ i χ α α (n) is a ( α)00% confidence interval for the parameter.

9 3. Let X, X,..., X n be a random sample from a population with Pareto density β x (+) for β x < f(x;, α) = where is an unknown parameter and β > 0 is a known parameter. Show that ( n i= ln χ α (n) ) X i β is a ( α)00% confidence interval for., n i= ln ( χ α (n) ) X i β 4. Let X, X,..., X n be a random sample from a population with Laplace density where is an unknown parameter. Show that e x, < x < [ n i= X i (n), χ α is a ( α)00% confidence interval for. ] n i= X i χ α (n) 5. Let X, X,..., X n be a random sample from a population with density x 3 e x for 0 < x < where is an unknown parameter. Show that [ n i= X i χ α (n), is a ( α)00% confidence interval for. n i= X i χ α (n) 6. Let X, X,..., X n be a random sample from a population with density x β β for 0 < x < (+x f(x;, β) = β ) + ]

10 where is an unknown parameter and β > 0 is a known parameter. Show that χ α (n) n i= ( ln + X β i ), is a ( α)00% confidence interval for. χ α (n) n i= ln ( + X β i ) 7. If X, X,..., X n is a random sample from a population with density π e (x ) if x < where is an unknown parameter, what is a ( α)00% approximate confidence interval for if the sample size is large? 8. Let X, X,..., X n be a random sample of size n from a distribution with a probability density ( + ) x if < x < where 0 < is a parameter. What is a ( α)00% approximate confidence interval for if the sample size is large? 9. Let X, X,..., X n be a random sample of size n from a distribution with a probability density x e x if 0 < x < where 0 < is a parameter. What is a ( α)00% approximate confidence interval for if the sample size is large? 0. Let X, X,..., X n be a random sample from a distribution with density β e (x 4) β for x > 4 where β > 0. What is a ( α)00% approximate confidence interval for if the sample size is large?

11 GROUP B. Five trials X, X,..., X 5 of a Bernoulli experiment were conducted to test H o : p = against H a : p = 3 4. The null hypothesis H o will be rejected if 5 i= X i = 5. Find the probability of Type I and Type II errors.. A manufacturer of car batteries claims that the life of his batteries is approximately normally distributed with a standard deviation equal to 0.9 year. If a random sample of 0 of these batteries has a standard deviation of. years, do you think that σ > 0.9 year? Use a 0.05 level of significance. 3. Let X, X,..., X 8 be a random sample of size 8 from a Poisson distribution with parameter λ. Reject the null hypothesis H o : λ = 0.5 is the observed sum 8 i= x i 8. First, compute the significance level α of the test. Second, find the power β(λ) of the test as a sum of Poisson probabilities when H a is true. 4. Suppose X has the density { for 0 < x < 0 otherwise. If one observation of X is taken, what are the probabilities of Type I and Type II errors in testing the null hypothesis H o : = against the alternative hypothesis H a : =, if H o is rejected for X > Let X have the density { ( + ) x for 0 < x < where > 0 0 otherwise. The hypothesis H o : = is to be rejected in favor of H : = if X > What is the probability of Type I error? 6. Let X, X,..., X 6 be a random sample from a distribution with density { () x for 0 < x < where > 0 0 otherwise.

12 The null hypothesis H o : = is to be rejected in favor of the alternative H a : > if and only if at least 5 of the sample observations are larger than 0.7. What is the significance level of the test? 7. A researcher wants to test H o : = 0 versus H a : =, where is a parameter of a population of interest. The statistic W, based on a random sample of the population, is used to test the hypothesis. Suppose that under H o, W has a normal distribution with mean 0 and variance, and under H a, W has a normal distribution with mean 4 and variance. If H o is rejected when W >.50, then what are the probabilities of a Type I or Type II error respectively? 8. Let X and X be a random sample of size from a normal distribution N(µ, ). Find the best likelihood ratio critical region of size for testing the null hypothesis H o : µ = 0 against the composite alternative H a : µ 0? 9. Suppose X, X,..., X 0 be a random sample from a Poisson distribution with mean. What is the most powerful (or best ) critical region of size 0.08 for testing the null hypothesis H 0 : = 0. against H a : = 0.5? 0. Let X be a random sample of size from a distribution with probability density f(x, ) = { ( ) + x if 0 x 0 otherwise. For a significance level α = 0., what is the best critical region for testing the null hypothesis H o : = against H a : =? GROUP C. Let X, X be a random sample of size from a distribution with probability density x e x! if x = 0,,, 3,... f(x, ) = where 0. For a significance level α = 0.053, what is the best critical region for testing the null hypothesis H o : = against H a : =? Sketch the graph of the best critical region.

13 . Let X, X,..., X 8 be a random sample of size 8 from a distribution with probability density x e x! if x = 0,,, 3,... f(x, ) = where 0. What is the best likelihood ratio critical region for testing the null hypothesis H o : = against H a :? If α = 0. can you determine the best likelihood ratio critical region? 3. Let X, X,..., X n be a random sample of size n from a distribution with probability density x 6 e β x Γ(7)β, if x > 0 f(x, ) = 7 where 0. What is the best likelihood ratio critical region for testing the null hypothesis H o : β = 5 against H a : β 5? What is the most powerful test? 4. Let X, X,..., X 5 denote a random sample of size 5 from a population X with probability density ( ) x if x =,, 3,..., where 0 < < is a parameter. What is the likelihood ratio critical region for testing H o : = 0.5 versus H a : 0.5? 5. Let X, X, X 3 denote a random sample of size 3 from a population X with probability density f(x; µ) = π e (x µ) < x <, where < µ < is a parameter. What is the likelihood ratio critical region for testing H o : µ = 3 versus H a : µ 3? 6. Let X, X, X 3 denote a random sample of size 3 from a population X with probability density e x if 0 < x <

14 where 0 < < is a parameter. What is the likelihood ratio critical region for testing H o : = 3 versus H a : 3? 7. Let X, X, X 3 denote a random sample of size 3 from a population X with probability density e x x! if x = 0,,, 3,..., where 0 < < is a parameter. What is the likelihood ratio critical region for testing H o : = 0. versus H a : 0.? 8. A box contains 4 marbles, of which are white and the rest are black. A sample of size is drawn to test H o : = versus H a :. If the null hypothesis is rejected if both marbles are the same color, find the significance level of the test. 9. Let X, X, X 3 denote a random sample of size 3 from a population X with probability density for 0 x where 0 < < is a parameter. What is the likelihood ratio critical region of size 7 5 for testing H o : = 5 versus H a : 5? 30. Let X, X and X 3 denote three independent observations from a distribution with density x β e β for 0 < x < f(x; β) = where 0 < β < is a parameter. What is the best critical region of size 0.05 for testing H o : β = 5 versus H a : β = 0?

15 Math 56 Homework 4 November, 006 Dr. Ron Sahoo Imagination is more important than knowledge. Albert Einstein Direction: This homework worths 0 points and is due on November 30, 006. In order to receive full credit, answer each problem completely and must show all work.. The following data were obtained from the grades of six students selected at random: Mathematics Grade, x English Grade, y Find the sample correlation coefficient r and then test the null hypothesis H o : ρ = 0 against the alternative hypothesis H a : ρ 0 at a significance level The data on the heights of 4 infants are: 8.,.4, 6.7 and 3.. For a significance level α = 0., use Kolmogorov-Smirnov Test to test the hypothesis that the data came from some uniform population on the interval (5, 5). (Use d 4 = 0.56 at α = 0..) 3. Test at the 0% significance level the hypothesis that the following data give the values of a random sample of size 50 from an exponential distribution with probability density where > 0. e x if 0 < x < 0 elsewhere,

Direction: This test is worth 250 points and each problem worth points. DO ANY SIX

Term Test 3 December 5, 2003 Name Math 52 Student Number Direction: This test is worth 250 points and each problem worth 4 points DO ANY SIX PROBLEMS You are required to complete this test within 50 minutes