= Inverse-variance weighted mean
{title2=$\widehat\mu=\sum_iw_iX_i$}
For independent <unbiased estimators> $X_i$ of a common mean with known positive <variances> $v_i$, the <linear unbiased estimator> with smallest variance has weights $w_i=v_i^{-1}/\sum_jv_j^{-1}$ and variance $(\sum_jv_j^{-1})^{-1}$. Unbiasedness requires $\sum_iw_i=1$. Minimizing $\sum_iv_iw_i^2$ by a <Lagrange multiplier> gives the weights, and the strictly positive diagonal <Hessian matrix> proves a unique global minimum. This conclusion requires no normality assumption.
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