Uniform Distribution

Abstract

Uniform distribution is the simplest statistical distribution. Although there is hardly any hydrologic variable that follows a uniform probability distribution, it is invoked in a variety of applications. For example, in Bayesian statistical modeling in hydrology it is frequently used as a prior distribution. In systems hydrology, uniform distribution is the pulse function obtained by subtracting two step functions lagged by the length of the uniform distribution. The pulse function is a key to deriving the unit hydrograph theory. The instantaneous unit hydrograph of the rational method, used in urban hydrology, is a uniform distribution (Singh, 1988). Of all the statistical distributions, uniform distribution has the highest entropy. In river morphology, when a river approaches equilibrium or dynamic equilibrium, its characteristics tend to follow a uniform distribution. Under equilibrium, rivers follow the minimum rate of energy dissipation. Furthermore, a river constantly adjusts its cross-sectional geometry and longitudinal profile to accommodate the influx of water and sediment coming from its drainage basin, and this adjustment is in accordance with the principle of maximum entropy. Thus, there is a close link between equilibrium and uniform distribution and then between maximum entropy (uniform distribution) and minimum rate of energy dissipation. This link plays a fundamental role in river engineering and training works, river morphology, evolution of deltas, etc.