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Assume that some random variable x is normally distributed with mean μ and variance σ². consider the following estimator where ci are constant values.

x-star =∑cixi

1. What is the expected value of x-star?

2. What is the least restrictive assumption that you can make about the ci such that x-star is an unbiased estimator of the population mean μ?

3. Assuming the covariances between all the different values of x in the sample are zero, what is the variance of x-star?

4. Assume that your condition of part B holds. Can we choose between x-bar (defined in the usual way) and x-star on the basis of unbiasedness? Why?

5. Use Chebychev's inequality to show why we prefer x-bar to x-star.

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1. E(x-star) =E(∑cixi)

=E(ci∑xi)

=ci∑E(xi)

=ci∑μ

=cinμ

2. I do not know what to do here. If x-star is an unbiased estimator of μ,

then E(x-star) = μ

if ci = 1/n then E(x-star) = μ, but that would make x-star equal to x-bar defined in the usual way: x-bar = 1/n∑xi.

3. var(x-star)=var(∑cixi)

=ci²var(∑xi)

=ci²(∑var(xi)+∑∑cov(xi,xj))

=ci²∑var(xi)