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.***Note: Questions B and C are worded slightly different from similar problem showing in Chegg. So, please do not copy!

a) It is given that, the independent random variables X1, X2, X3 with mean µ and variance σ2.

Answered: Consider the following weighted…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter1: Functions

Section1.2: The Least Square Line

Problem 5E

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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.

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

***Note: Questions B and C are worded slightly different from similar problem showing in Chegg. So, please do not copy!

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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