consider a random variable $X$, whose mean value is known and stochastically independent repetitions $X_1,...,X_n$ of $X$.
For each vector $\overrightarrow{a}=(a_1,...,a_n) \in \mathbb{R}_{n} \text{ with } a_i > 0 $ we denote with $T_\overrightarrow{a}$ the estimator
$$T_\overrightarrow{a} = a_1 . (X_1-\mu)^2 + a_2 . (X_2-\mu)^2 + ... + a_n . (X_n-\mu)^2$$ for the variance of X.
a) Determine all $\overrightarrow{a}$ for which $T_\overrightarrow{a}$ is an unbiased estimator for the variance $Var(X)$ of $X$.
b) Determine the most effective among the unbiased estimators $T_\overrightarrow{a}$.
My thoughts
a) $$Var(T_\overrightarrow{a} ) = a_1^2 . Var((X_1-\mu)^2) + a_2^2 . Var((X_2-\mu)^2) + ... + a_n^2 . Var((X_n-\mu)^2) \\ = a_1^2 . Var((X-\mu)^2) + a_2^2 . Var((X-\mu)^2) + ... + a_n^2 . Var((X-\mu)^2) \\ =(a_1^2 + a_2^2 + ... + a_n^2) . Var((X-\mu)^2) \\ $$
I know, how to determine all $\overrightarrow{a}$ for which $T_\overrightarrow{a}$ is an unbiased estimator for the expected value $E(X)$ of $X$, but how can I do the same variance $Var(X)$ of $X$?
Every advice to solve this problem would be appreciated. Thanks :-)