Probability Distribution Density of Uncertainty, Uncertainty Region and Uncertainty Relation

We study the distribution behaviors of uncertainties, as quantifiedby the standard deviation (square root of variance), of quantumobservables in random quantum states (in general mixed) induced fromthe uniform Haar measure on the pure states of the purified system.The joint probability distribution densities of uncertainties ofqubit quantum observables are derived analytically. This opens a newvista for studying uncertainty relations in a more detailed andanalytical way. The results maybe be generalized to multipleobservables in higher dimensional spaces.