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Independent observations X₁, X₂ and X, come from normal distributions that have the same mean, μ, but whose variances are 3,2 and 4 respectively, so that X₁~ N(1, 3), X₂~ N(μ, 2), X₂~ N(µ,4). One possible estimator of μ is P₁ = (2X₁ +5X₂+4Xg). (a) Show that is an unbiased estimator of . (b) What is the variance of ? Carefully explain your workings and reasonings. (c) The estimator A₂ = (X₁ + X₂ + X₂) is also an unbiased estimator of μ. Given that the variance of ₂ is 1, which of the estimators , and 2, is the better estimator of μ? Justify your answer.

Independent observations X₁, X₂ and X, come from normal distributions that have the same mean, μ, but whose variances are 3,2 and 4 respectively, so that X₁~ N(1, 3), X₂~ N(μ, 2), X₂~ N(µ,4). One possible estimator of μ is P₁ = (2X₁ +5X₂+4Xg). (a) Show that is an unbiased estimator of . (b) What is the variance of ? Carefully explain your workings and reasonings. (c) The estimator A₂ = (X₁ + X₂ + X₂) is also an unbiased estimator of μ. Given that the variance of ₂ is 1, which of the estimators , and 2, is the better estimator of μ? Justify your answer.

MATLAB: An Introduction with Applications
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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Independent observations X₁, X₂ and X3 come from normal distributions that have the same mean, μ, but whose variances are 3,2 and 4 respectively, so that X₁ ~ N(3), X₂ ~ N(µ, 2), X₂ ~ N(1, 2), X3~ N(µ,4). One possible estimator of μ is P₁ = (2X₁ +5X₂ + 4Xg). (a) Show that is an unbiased estimator of μ. (b) What is the variance of ? Carefully explain your workings and reasonings. (c) The estimator A2 = (X₁ + X₂ + X₂) is also an unbiased estimator of u. Given that the variance of ₂ is 1, which of the estimators and 2, is the better estimator of μ? Justify your answer.
Transcribed Image Text:Independent observations X₁, X₂ and X3 come from normal distributions that have the same mean, μ, but whose variances are 3,2 and 4 respectively, so that X₁ ~ N(3), X₂ ~ N(µ, 2), X₂ ~ N(1, 2), X3~ N(µ,4). One possible estimator of μ is P₁ = (2X₁ +5X₂ + 4Xg). (a) Show that is an unbiased estimator of μ. (b) What is the variance of ? Carefully explain your workings and reasonings. (c) The estimator A2 = (X₁ + X₂ + X₂) is also an unbiased estimator of u. Given that the variance of ₂ is 1, which of the estimators and 2, is the better estimator of μ? Justify your answer.
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