  Recently, Maccone and Pati have given two stronger uncertainty relations
based on the sum of variances and one of them is nontrivial when the quantum
state is not an eigenstate of the sum of the observables. We derive a family of
weighted uncertainty relations to provide an optimal lower bound for all
situations and remove the restriction on the quantum state. Generalization to
multi-observable cases is also given and an optimal lower bound for the
weighted sum of the variances is obtained in general quantum situation.
