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Consider the following weighted averages of independent random variables X1, X2 , X3 all with mean µ and variance σ2 : ˆθ1 = 1/3 X1 + 1/3 X2 + 1/3 X3 ˆθ2 = 1/4 X1 + 2/4 X2 + 1/4 X3 ˆθ3 = 2/5 X1 + 2/5 X2 + 2/5 X3 (a) Find E[ ˆθ1], E[ ˆθ2], and E[ ˆθ3] (b) Are ˆθ1, ˆθ1, and ˆθ1 unbiased for µ? Please Explain. (c) Find the variance for ˆθ1, ˆθ1, and ˆθ1 (d) If you had to use one of the above unbiased estimators, which would you pick? Explain.

Consider the following weighted averages of independent random variables X1, X2 , X3 all with mean µ and variance σ2 : ˆθ1 = 1/3 X1 + 1/3 X2 + 1/3 X3 ˆθ2 = 1/4 X1 + 2/4 X2 + 1/4 X3 ˆθ3 = 2/5 X1 + 2/5 X2 + 2/5 X3 (a) Find E[ ˆθ1], E[ ˆθ2], and E[ ˆθ3] (b) Are ˆθ1, ˆθ1, and ˆθ1 unbiased for µ? Please Explain. (c) Find the variance for ˆθ1, ˆθ1, and ˆθ1 (d) If you had to use one of the above unbiased estimators, which would you pick? Explain.

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
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Consider the following weighted averages of independent random variables X1, X2 , X3 all with mean µ and variance σ2 :

ˆθ1 = 1/3 X1 + 1/3 X2 + 1/3 X3

ˆθ2 = 1/4 X1 + 2/4 X2 + 1/4 X3

ˆθ3 = 2/5 X1 + 2/5 X2 + 2/5 X3

(a) Find E[ ˆθ1], E[ ˆθ2], and E[ ˆθ3]

(b) Are ˆθ1, ˆθ1, and ˆθ1 unbiased for µ? Please Explain.

(c) Find the variance for ˆθ1, ˆθ1, and ˆθ1

(d) If you had to use one of the above unbiased estimators, which would you pick? Explain.

Expert Solution
Step 1

Introduction:

The random variables X1, X2 , X3 are selected independently from a distribution with mean µ and variance σ2.

Thus,

EX1=EX2=EX3=μ;

VX1=VX2=VX3=σ2.

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