6. Consider the model Y₁ = B₁x;+Єi, i = 1, . . ., n, with mean 0 and variance σ². where €1, , εn are independent (a). Derive B₁, the least squares estimator of ß₁. (b). Derive E[1] and Var(B1). You may state (without proof) and use expressions for the expectation and variance of a linear combination of random variables. (c). If E[Y] at a new x value, xnew, is estimated by Ŷ new = Xnewẞ1, write down the variance of Ŷnew. (d). For what values of new is the variance in part (c) minimized? Does this make sense?
6. Consider the model Y₁ = B₁x;+Єi, i = 1, . . ., n, with mean 0 and variance σ². where €1, , εn are independent (a). Derive B₁, the least squares estimator of ß₁. (b). Derive E[1] and Var(B1). You may state (without proof) and use expressions for the expectation and variance of a linear combination of random variables. (c). If E[Y] at a new x value, xnew, is estimated by Ŷ new = Xnewẞ1, write down the variance of Ŷnew. (d). For what values of new is the variance in part (c) minimized? Does this make sense?
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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Refer to the picture for the question, and show all work neccessary in full detail
![6.
Consider the model Y₁ = B₁x;+Єi, i = 1, . . ., n,
with mean 0 and variance σ².
where €1,
, εn are independent
(a). Derive B₁, the least squares estimator of ß₁.
(b). Derive E[1] and Var(B1). You may state (without proof) and use expressions for the
expectation and variance of a linear combination of random variables.
(c). If E[Y] at a new x value, xnew, is estimated by Ŷ new = Xnewẞ1, write down the variance
of Ŷnew.
(d). For what values of new is the variance in part (c) minimized? Does this make sense?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F643c9073-bfc9-41d3-b50c-3b14187fea75%2F929ad046-8b11-4b1d-8a5e-f0739be66fdd%2F3badhnw_processed.png&w=3840&q=75)
Transcribed Image Text:6.
Consider the model Y₁ = B₁x;+Єi, i = 1, . . ., n,
with mean 0 and variance σ².
where €1,
, εn are independent
(a). Derive B₁, the least squares estimator of ß₁.
(b). Derive E[1] and Var(B1). You may state (without proof) and use expressions for the
expectation and variance of a linear combination of random variables.
(c). If E[Y] at a new x value, xnew, is estimated by Ŷ new = Xnewẞ1, write down the variance
of Ŷnew.
(d). For what values of new is the variance in part (c) minimized? Does this make sense?
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