Example 9.5: Let X1, X2, X3, and X4 be a random sample from a distribution with mean and variance o². Consider the following four estimators of u μ = X₁ μ = 0.1X₁ +0.2X₂ +0.3X3 +0.4X4 = X₂ + X₂ 2 a) Show that all four estimators are unbiased. î₂ M₁ = X wodisiyoh bisbunte, alte & notariteo na to sovy k rysb bilinea con i bebine T-samling 20bies & Mapigut erabide biobrate bonites ne fa nomogong siqace di bas 7. nom siquies di to atons bisher sill sandW:0. stqmax3 b) Calculate the variance of each estimator. Which one has the smallest variance?
Example 9.5: Let X1, X2, X3, and X4 be a random sample from a distribution with mean and variance o². Consider the following four estimators of u μ = X₁ μ = 0.1X₁ +0.2X₂ +0.3X3 +0.4X4 = X₂ + X₂ 2 a) Show that all four estimators are unbiased. î₂ M₁ = X wodisiyoh bisbunte, alte & notariteo na to sovy k rysb bilinea con i bebine T-samling 20bies & Mapigut erabide biobrate bonites ne fa nomogong siqace di bas 7. nom siquies di to atons bisher sill sandW:0. stqmax3 b) Calculate the variance of each estimator. Which one has the smallest variance?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![Example 9.5: Let \( X_1, X_2, X_3, \) and \( X_4 \) be a random sample from a distribution with mean \( \mu \) and variance \( \sigma^2 \). Consider the following four estimators of \( \mu \):
\[
\hat{\mu}_1 = X_1 \quad \hat{\mu}_2 = \frac{X_2 + X_3}{2} \quad \hat{\mu}_3 = 0.1X_1 + 0.2X_2 + 0.3X_3 + 0.4X_4 \quad \hat{\mu}_4 = \bar{X}
\]
a) Show that all four estimators are unbiased.
b) Calculate the variance of each estimator. Which one has the smallest variance?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F243a7230-816a-40d0-9f50-4da38fdaa0a6%2F20875b7c-732c-4e0f-a72a-784479b82f10%2Fu5yjcz5h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Example 9.5: Let \( X_1, X_2, X_3, \) and \( X_4 \) be a random sample from a distribution with mean \( \mu \) and variance \( \sigma^2 \). Consider the following four estimators of \( \mu \):
\[
\hat{\mu}_1 = X_1 \quad \hat{\mu}_2 = \frac{X_2 + X_3}{2} \quad \hat{\mu}_3 = 0.1X_1 + 0.2X_2 + 0.3X_3 + 0.4X_4 \quad \hat{\mu}_4 = \bar{X}
\]
a) Show that all four estimators are unbiased.
b) Calculate the variance of each estimator. Which one has the smallest variance?
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