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

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Consider a population with mean μ and variance σ2<∞. You are comparing 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)=9; E( ˆ μ 1)=μ+1;V( ˆ μ 2)=1. We also know that the covariance between the two estimators is COV( ˆ μ 1, ˆ μ 2)=−2. 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

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

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