Using a suite of self-similar cosmological simulations, we measure the probability distribution functions (PDFs) real-space density, redshift-space and their geometric mean. We find that density PDF is well-described by function two parameters: $n_s$, spectral slope, $\sigma_L$, linear rms fluctuation. For mean real- densities, introduce third parameter, $s_L={\sqrt{\langle(dv^L_{\rm pec}/dr)^2...

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 ...