1) This question considers different unbiased estimators for the mean

1) This question considers different unbiased estimators for the mean.

(a) Let X₁, X₂, X₃ be independent random variables with E(X₁) = E(X₂) = E(X₃) = m and Var(X₁) = Var(X₂) = Var(X₃) = s². Let

           Y = X₁ + 2X₂ + 3X₃/6

Show that Y is an unbiased estimator for the mean m.

(b) For the estimator Y from part (a), show that Var(Y ) > Var(X), where X is the mean of the X_i. Comment on this result.

(c) Now consider the case where we have n random variables, X₁,....,X_n which are independent with E(X_i) = m and Var(X_i) = s² for all i €{1,2,...,n}. For c₁,...,c_n eR, define

  Z = Sⁿ_i=1c_iX_i

Determine the variance of Z.

(d) Which condition on ¢₁,...,c_n ensures that Z from part (c) is an unbiased estimator for m?

(e) Assume that the c_j are such that Z from part (c) is an unbiased estimator for m. Using the Cauchy-Schwarz inequality, or otherwise, show that Var(Z) ³ Var(X ), where X is the mean of X₁,..,X_n.