Let x1 and x2 be independent random variables with mean m and varian.pdf

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Let x1 and x2 be independent random variables with mean m and variance rho^2. Suppose that we have two estimators of m: theta hat 1=(x1+x2)/2 and theta hat 2 = (x1+3x2)/4 Are both estimators unbiased estimators of m? What is the variance of each estimator? Solution For theta hat 1= (x1+x2)/2 Expectation= (1/2)*(m+m)=m so this estimator is unbiased Variance= (1/4)*(2rho^2)=(1/2)*rho^2 because x1 and x2 are independent. For theta hat 2=(x1+3x2)/4 Expectation=(1/4)*(m+3m)=m so this estimator is unbiased. Variance=(1/16)*(rho^2+9rho^2)=(10/16)*rho^2=(5/8)*rho^2 We observe that the variance of the first estimator is smaller than that of the second so we would prefer the first estimator theta hat 1..

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