Your Textbook States That an Implication of the Gauss-Markov Theorem $\bar { Y }$

Essay

Your textbook states that an implication of the Gauss-Markov theorem is that the sample average, $\bar { Y }$ , is the most efficient linear estimator of $E \left( Y _ { i } \right)$ when $Y _ { 1 } , \ldots , Y _ { n }$ are i.i.d. with $E \left( Y _ { i } \right) = \mu _ { Y }$ and $\operatorname { var } \left( Y _ { i } \right) = \sigma _ { Y } ^ { 2 }$ . This follows from the regression model with no slope and the fact that the OLS estimator is BLUE.
Provide a proof by assuming a linear estimator in the $Y$ 's, $\tilde { \mu } = \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i }$ . (a)State the condition under which this estimator is unbiased.