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Let Y,, Y2, Y3, and Y, be independent, identically distributed random variables from a population with mean u and variance o?. Let Y = -(Y, + Y, + Y, + Y4) denote the average of these four random variables. (i) What are the expected value and variance of Y in terms of u and o²? (ii) Now, consider a different estimator of µ: W = -Y, Y, + *A + 1? + + This is an example of a weighted average of the Y,. Show that W is also an unbiased estimator of u. Find the variance of W. (iii) Based on your answers to parts (i) and (ii), which estimator of m do you prefer, Y or W?

Let Y,, Y2, Y3, and Y, be independent, identically distributed random variables from a population with mean u and variance o?. Let Y = -(Y, + Y, + Y, + Y4) denote the average of these four random variables. (i) What are the expected value and variance of Y in terms of u and o²? (ii) Now, consider a different estimator of µ: W = -Y, Y, + *A + 1? + + This is an example of a weighted average of the Y,. Show that W is also an unbiased estimator of u. Find the variance of W. (iii) Based on your answers to parts (i) and (ii), which estimator of m do you prefer, Y or W?

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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Let Y,, Y2, Y3, and Y, be independent, identically distributed random variables from a population with mean u and variance o?. Let Y = -(Y, + Y, + Y, + Y4) denote the average of these four random variables. (i) What are the expected value and variance of Y in terms of u and o²? (ii) Now, consider a different estimator of µ: W = -Y, Y, + *A + 1? + + This is an example of a weighted average of the Y,. Show that W is also an unbiased estimator of u. Find the variance of W. (iii) Based on your answers to parts (i) and (ii), which estimator of m do you prefer, Y or W?
Transcribed Image Text:Let Y,, Y2, Y3, and Y, be independent, identically distributed random variables from a population with mean u and variance o?. Let Y = -(Y, + Y, + Y, + Y4) denote the average of these four random variables. (i) What are the expected value and variance of Y in terms of u and o²? (ii) Now, consider a different estimator of µ: W = -Y, Y, + *A + 1? + + This is an example of a weighted average of the Y,. Show that W is also an unbiased estimator of u. Find the variance of W. (iii) Based on your answers to parts (i) and (ii), which estimator of m do you prefer, Y or W?
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