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Consider the simple regression model Y = a +3X + ε where ~ N(0,0²), for unknown parameters a, ß, o². Let â, ß, ô² 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 ß := Â(Y1, Y2, · Yn) Var (3) - = n 1 1 —— Vx n 1 0² Σ(x¡ — ñ)Yį. - 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 Eß in lecture.)

Consider the simple regression model Y = a +3X + ε where ~ N(0,0²), for unknown parameters a, ß, o². Let â, ß, ô² 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 ß := Â(Y1, Y2, · Yn) Var (3) - = n 1 1 —— Vx n 1 0² Σ(x¡ — ñ)Yį. - 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 Eß in lecture.)

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 (i) and (ii)? Thank you so much for your help

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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Consider the simple regression model :

Y=α+βX+ε

where ε~N0, σ2, for unknown parameters α, β, σ2

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