The equations for the higher-order moments and probability density function of turbulent velocity fluctuations are considered. These are derived utilizing the basic hydrodynamic equations of fluid flow. Using truncated cumulant expansions as approximations for the probability density distributions of the corresponding turbulence quantities, an alternative set of equations for the moments is formulated that contains only velocity correlations. From these equations, interrelations between the higher-order moments are deduced. Several theoretically derived relationships between correlations of different orders are experimentally verified using data available in the literature and also data measured by the authors. In the paper, an attempt is made to reconstruct the entire probability density distributions from derived inter-relations between the higher-order moments.