(Sec. 6.1) Using a long rod that has length \( \mu \) (unknown), you are going to lay out a square plot in which the length of each side is \( \mu \). Thus the area of the plot will be \( \mu^2 \). However, because you do not know the value of \( \mu \), you decide to make \( n \) independent measurements \( X_1, \ldots, X_n \) of the length. Assume that each \( X_i \) has mean \( \mu \) and variance \( \sigma^2 \).(a) Show that \( \bar{X}^2 \) is not an unbiased estimator for the area of the square plot \( \mu^2 \). [Hint: for any random variable \( Y \), \( E[Y^2] = V[Y] + E[Y]^2 \). Apply this for \( Y = \bar{X} \).](b) For what value of \( k \) is the estimator \( \bar{X}^2 - kS^2 \) unbiased for \( \mu^2 \)?

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Description: (Sec. 6.1) Using a long rod that has length \( \mu \) (unknown), you are going to lay out a square plot in which the length of each side is \( \mu \). Thus the area of the plot will be \( \mu^2 \). However, because you do not know the value of \( \m

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(a) Show that X² is not an unbiased estimator for the area of the square plot p². [Hint: for any rv Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.] (b) For what value of k is the estimator X² – kS² unbiased for µ²?

(a) Show that X² is not an unbiased estimator for the area of the square plot p². [Hint: for any rv Y, E[Y²] = V[Y] + E[Y]². Apply this for Y = X.] (b) For what value of k is the estimator X² – kS² unbiased for µ²?

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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