Consider again a random sample X1,..., Xn from the population distribution withdensityf (x; 0) = 0xº-1, x € (0, 1),and parameter 0 E (0, 00), as in Q2.(1) Is the above parametric model an exponential family? Justify your answer.(2) Is there an unbiased estimator of 0 whose variance reaches the Cramér-Raolower bound? Justify your answer.(3) Show that T = -(1/n) E log(X;) is sufficient for 0.

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Answered: Consider again a random sample X1,...,…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Book Pages (x) Price (y) A 500 $7.10 B 700 7.60 C 750 9.10 D 590 6.60 E 560 7.60 F 650 7.10 G 475 5.10 (a) Develop a least squares estimated regression line. (Round your numerical values to three decimal places.) (b) Compute the coefficient of determination. (Round your answer to four decimal places.) Explain its meaning. (Give your answer as a percent. Round your answer to two decimal places.) The value of the coefficient of determination tells us that % of the variability in |--?-- ☑ has been explained by the least squares regression equation.
(16) The moment-generating function of the geometric random variable X with parameter p is M(t) = 1-per. Use this to find the mean and variance of X.
Let X₁,..., Xn be a random sample from a distribution with one of two pdfs. If 0 = 1, then f(x;0=1) = 1 (0 < x < 1). If 0 = 2, then f(x;0= 2) = 2x1 (0 < x < 1). (a) Give a general form of the MLE of 0. (b) You are given a sample data of 3 values: 0.4, 0.5, 0.8. Find the MLE estimate of 0 based on the data.
Suppose that the response y is generated by y = f(x) + €, where e is a zero-mean Gaussian noise with variance 1. a) Suppose that f(x) = x. Randomly generate 10 x's and generate the corresponding y's; you need to generate two random numbers (i.e., x and e for each of the 10 points). Fit the data with linear regression and plot the scatter points. b) Suppose that f(x) = x². Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points. c) Suppose that f(x) = 1/x. Randomly generate 10 (x, y) pairs. Fit the data with linear regression and plot the scatter points.
6. Show that 1 s2 E1(Xi – x)² is unbiased estimator of the population variance o? i=1 п-1
#2: Suppose a random variable X has expected value E[X] = 3 and variance Var(X) = 4. Compute the following quantities. (a) E[3X + 2] (b) E[X²] (c) E[(3X+2)²] (d) Var[2X + 2]
2.2 Losses at an oil refinery (in millions of dollars) due to excess heat can be divided according to the reason behind the generation of excessive heat. oversupplying fuel excess air carelessness of operator incomplete combustion 202 124 96 27 (a) Draw a Pareto chart. (b) What percent of the loss occurs due to (1) excess air? (2) excess air and oversupplying fuel? 2.7
Let X1, X2, ..., X, be a random sample from f(x;0) = 0e-tI(0,00)(x). Find a 100y or (1- a)100 percent confidence interval for the mean of the population.
Let x1, x2, ..., n represent a random sample from a distribution with mean E(x) and variance Var(x). Show that Cov(x, x₁ - x) = 0.
The information associated with the maximum likelihood estimator of a parameter 0 is 4n, where n is the number of observations. Calculate the asymptotic variance of the maximum likelihood estimator of 20 . (A) (В) (C) (D) 8n (E) 16n
Suppose we collect n samplesX1,X2, . . . ,Xn from a uniform distribution U[θ,1]. We know the maximum likelihood estimator of θ is θ = min(Xi). (1) What is the CDF of θ? (2) What is the pdf of θ? (3) Find the bias of θ. (4) Find the variance of θ. (5) Find the MSE of θ.
This problem explores the question of estimating o(> 0) using a confidence interval based on a random sample X1, X2, ..., Xn from the distribution with pdf fo(x) = {& exp(- 20 if x > 0 otherwise. (a) Find the pdf of X2 by finding the cdf and differentiating it. (b) Using the result of (a) and techniques we developed in class, propose a pivot for estimating using a confidence interval and find the distribution of your pivot. Review Slides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is used to construct a confidence interval. (c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula for finding a short 100(1-a)% confidence interval for o. (d) Using the formula obtained in (c), compute the width of the resulting 95% confidence interval when n = 1 and ₁ = 1. (e) Is the 95% confidence interval for a you obtained in (d) when n = 1 and ₁ = 1 the shortest possible? If your answer is yes, explain why. If the answer is no, give a…

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