(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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(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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