(3) Suppose you have an independent sample of two observations, denoted y₁ and y2, from a population of interest. Further, suppose that E(yi) = μ and Var(yi) = o², i = 1, 2. Consider the following estimator of μ: û = cy₁ + dy2, for some given constants c and d that you are able to choose. Think about this question as deciding how to weight the observations y₁ and y2 (by choosing c and d) when estimating μ. (3a) Under what condition will be an unbiased estimator of ? (Your answer will state a restriction on the constants c and d in order for the estimator to be unbiased). (3b) Given your answer in (3a), solve for d in terms of c and substitute that result back into the expression for above. Note that the resulting estimator, now a function of c only, is unbiased. Once you have made this substitution, what is the variance of û in terms of o² and c? 2 3 (3c) What is the value of c that minimizes the variance expression in (3b)? (You will need to take a derivative to determine the optimal c). Can you provide any intuition for this result?

(3) Suppose you have an independent sample of two observations, denoted y₁ and y2, from a population of interest. Further, suppose that E(yi) = μ and Var(yi) = o², i = 1, 2. Consider the following estimator of μ: û = cy₁ + dy2, for some given constants c and d that you are able to choose. Think about this question as deciding how to weight the observations y₁ and y2 (by choosing c and d) when estimating μ. (3a) Under what condition will be an unbiased estimator of ? (Your answer will state a restriction on the constants c and d in order for the estimator to be unbiased). (3b) Given your answer in (3a), solve for d in terms of c and substitute that result back into the expression for above. Note that the resulting estimator, now a function of c only, is unbiased. Once you have made this substitution, what is the variance of û in terms of o² and c? 2 3 (3c) What is the value of c that minimizes the variance expression in (3b)? (You will need to take a derivative to determine the optimal c). Can you provide any intuition for this result?

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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