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Questions

BU.232.630.W4.SP25 sample_quiz_1

Single choice

Consider a population with mean μ and variance σ2<∞. Assume the following two estimators ˆ μ 1 and ˆ μ 2 for the mean of the population μ, with the following expected values and variances E( ˆ μ 1)=μ;V( ˆ μ 1)=5; E( ˆ μ 1)=μ+1;V( ˆ μ 2)=2. We also know that the covariance between the two estimators is COV( ˆ μ 1, ˆ μ 2)=−1. Now consider a new estimator that combines the two previous ones ˆ μ 3= 1 3 ˆ μ 1+ 2 3 ˆ μ 2. Then the variance V( ˆ μ 3) of ˆ μ 3 is

Options
A.1
B.3
C.σ2 + 1.4444
D.2.5556
E.1.4444
F.σ2 -1
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Step-by-Step Analysis
We are given two estimators for the mean μ with: E(μ̂1) = μ, Var(μ̂1) = 5; E(μ̂2) = μ + 1 (likely intended), Var(μ̂2) = 2; Cov(μ̂1, μ̂2) = -1. A new estimator is formed as μ̂3 = (1/3) μ̂1 + (2/3) μ̂2. To find Var(μ̂3), use Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X,Y). Comput......Login to view full explanation

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