  We study the uncertainties of quantum mechanical observables, quantified by
the standard deviation (square root of variance) in Haar-distributed random
pure states. We derive analytically the probability density functions (PDFs) of
the uncertainties of arbitrary qubit observables. Based on these PDFs, the
uncertainty regions of the observables are characterized by the supports of the
PDFs. The state-independent uncertainty relations are then transformed into the
optimization problems over uncertainty regions, which opens a new vista for
studying state independent uncertainty relations. Our results may be
generalized to multiple observable case in higher dimensional spaces.
