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(iv) Suppose that we know o² = 2 and are given x-values X1 2 X7 X2₂ X3 X4 X5 X6 3 4 5 6 7 8 What is the distribution of the MLE estimator ŝ? Compute the probability tha MLE estimate 3 differs from the true value of 3 by more than 0.1.

(iv) Suppose that we know o² = 2 and are given x-values X1 2 X7 X2₂ X3 X4 X5 X6 3 4 5 6 7 8 What is the distribution of the MLE estimator ŝ? Compute the probability tha MLE estimate 3 differs from the true value of 3 by more than 0.1.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Can you show me the answer to question 4 and 5? Thank you so much for your help

(iii) Using Theorem 5.5-1 from the textbook, conclude that 3 ~ N(3.17) -), Vx i.e. the estimator 3 follows a normal distribution with mean 6 and variance (iv) Suppose that we know o2 = 2 and are given x-values X1 X2 X3 X4 X5 X6 X7 2 3 4 5 6 7 8 1 0² V x N What is the distribution of the MLE estimator 3? Compute the probability that the MLE estimate differs from the true value of ß by more than 0.1.
Transcribed Image Text:(iii) Using Theorem 5.5-1 from the textbook, conclude that 3 ~ N(3.17) -), Vx i.e. the estimator 3 follows a normal distribution with mean 6 and variance (iv) Suppose that we know o2 = 2 and are given x-values X1 X2 X3 X4 X5 X6 X7 2 3 4 5 6 7 8 1 0² V x N What is the distribution of the MLE estimator 3? Compute the probability that the MLE estimate differs from the true value of ß by more than 0.1.
Consider the simple regression model Y = a + 3X + € where ~ N(0,0²), for unknown parameters a, 3, o². Let â, , ² be their MLE estima- tors, respectively, for sample (x₁, y₁), (x2, y2),, (xn, yn). In this exercise, we look at the probability distribution of the estimator 3 as a random variable. (i) Verify that the estimator as a random variable may be expressed as (ii) Show that 8:8(Y₁, Y₂, Yn) := ; = Var(3) = n 11 Vx n 10² Σ(x₁ - π) Yi. i=1 Vx n (Hint: use the "variances of independent RV's add up" property from Homework 2 Exercise 6 (ii). If you are not sure how to proceed, look at the computation of E3 in lecture.)
Transcribed Image Text:Consider the simple regression model Y = a + 3X + € where ~ N(0,0²), for unknown parameters a, 3, o². Let â, , ² be their MLE estima- tors, respectively, for sample (x₁, y₁), (x2, y2),, (xn, yn). In this exercise, we look at the probability distribution of the estimator 3 as a random variable. (i) Verify that the estimator as a random variable may be expressed as (ii) Show that 8:8(Y₁, Y₂, Yn) := ; = Var(3) = n 11 Vx n 10² Σ(x₁ - π) Yi. i=1 Vx n (Hint: use the "variances of independent RV's add up" property from Homework 2 Exercise 6 (ii). If you are not sure how to proceed, look at the computation of E3 in lecture.)
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