  The optimal state-independent lower bounds for the sum of variances or
deviations of observables are of significance for the growing number of
experiments that reach the uncertainty limited regime. We present a framework
for computing the tight uncertainty relations of variance or deviation via
determining the uncertainty regions, which are formed by the tuples of two or
more of quantum observables in random quantum states induced from the uniform
Haar measure on the purified states. From the analytical formulae of these
uncertainty regions, we present state-independent uncertainty inequalities
satisfied by the sum of variances or deviations of two, three and arbitrary
many observables, from which experimentally friend entanglement detection
criteria are derived for bipartite and tripartite systems.
