Let $X_1,...,X_n$ be uncorrelated random variables with $\mu \neq 0$ and $\sigma^2>0$.
Show that of all $T=\sum_i \alpha_iX_i$ with $\mathbb{E}(T) = \mu$ the sample mean $\bar{X}$ has the lowest variance.
I've already shown that $$Var(\bar{X})=\sigma^2/n$$ $$Var(T) = \sum_i \alpha_i^2 \sigma^2$$ But it seems that it depends on $\alpha_i$ what's smaller, doesn't it? But I have no information about $\alpha_i$. Am I missing something?
