Let $X_{1}, \ldots, X_{n}$ be a random sample from a distribution with mean $\mu$ and variance $\sigma^{2}$ Consider an estimator $T=\sum_{i=1}^{n} a_{i} X_{i}=\mathbf {a X}$, for $\mathbf{a}=\left(a_{1}, \ldots, ight.$, $\left.a_{n} ight)$ and $\mathbf{X}=\left(X_{1}, \ldots, x_{n} ight)^{T}$, so that a is a row vector and and $\mathbf{X}$ is a column vector. (a) Find a condition on a so that $T$ is an unbiased estimator of $\mu$. (b) Find $\mathbf {a}=\mathbf {a}_{0}$ so that the condition in part (a) is satisfied and $\operatorname{Var}(T)$ is a minimum. $T_{0}=\mathbf{a}_{0}^{T} \mathbf{X}$ is then the best linear unbiased estimator (the BLUE) of $\mu$. What is this minimum variance? SP.JG.136