题目
题目

BU.232.630.F3.SP25 sample quiz 1

单项选择题

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

选项
A.2.5556
B.3
C.1
D.σ2 + 1.4444
E.1.4444
F.σ2 -1
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标准答案
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思路分析
To evaluate the variance of the combined estimator μ̂3 = (1/3) μ̂1 + (2/3) μ̂2, we need to apply the variance formula for a linear combination of correlated estimators: Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y). Here, a = 1/3, b = 2/3, Var(μ̂1) = 5, Var(μ̂2) = 2, Cov(μ̂1, μ̂2) = -1. Now compute step by step: - a^2 Var(μ̂1) = (1/9) * 5 = 5/9. - b^2 Var(μ̂2) = (4/9) * 2 = 8/9. - 2ab Cov(μ̂1, μ̂2) = 2*(1/3)*(2/3)*(-1) = (4/9)*(-1)......Login to view full explanation

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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

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