Calculating Variance

Question

Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a population whose distribution is normal with mean $\mu$ and variance $\sigma^2$.

Consider the statistics $\displaystyle T_1 = \frac{X_1 − X_2 + X_3 + X_4 + X_5}{3}$ and $\displaystyle T_2 = \frac{X_1 +X_2 +2X_3 +X_4 +X_5}{6}$ as unbiased estimators for $\mu$. Find the statistic with the least variance.

Answer 1

Assuming covariances are all 0's, $\mathbf{Var} T_1 = \frac{5 \sigma^2}{9}$ and $\mathbf{Var} T_2 = \frac{2 \sigma^2}{9}$.

Answer 2

Hint: the variance of $\sum a_iX_i = a\cdot X$ is $$ a\cdot \left( \begin{array}{ccc} \sigma^2 & 0& 0&\dots & 0 \\ 0 & \sigma^2 & 0 & \dots & 0 \\ \vdots &&&&\vdots \\ 0 & 0 & 0&\dots & \sigma^2 \end{array} \right) a $$