Probability density functions of quantum mechanical observable uncertainties

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