A Simple Differential Equation Associating Multiple Types of Probability Distributions in Characterization of Noninear Processes in Geosciences

Abstract:

Probability distributions or probability density functions are commonly used in geosciences for purposes ranging from characterization, estimation, prediction and assessment. Each type of probability distribution is found suitable for describing certain types of phenomena and processes, for example, the normal distribution is used, ensured by the classical Central Limit Theorem, to characterizing physical quantities summed many independent processes such as measurement errors, gamma distribution is frequently used to model waiting times, Pareto distribution to extreme events, power-law distribution to frequency-size distribution of earthquakes and inverse gamma distribution for self-organized criticality description of landslides. Understanding the associations of these types of probability distributions is essential not only for determining usage of these distributions but also for interpretation of the results. In this paper we demonstrate that a simple non-linear first order differential equation can be used to describe the decay function of probability density around the mean values. Assume the decay rate of probability density function is negatively proportional to the density itself or a power of the density, with a functional coefficient dependent on the value of the random variable. Applying Taylor series expansion to the coefficient function, the differential equation can be approximated by multiple simple dynamic systems, each with explicit solutions. These functions can be utilized either as separate and combined solutions to generate various commonly used probability distributions including but not limited to Gaussian, power-law, gamma, inverse gamma, Pareto, Weibull, Rayleigh, and Maxwell–Boltzmann distributions. The association of these types of distributions provides insight into multiple types of probability distributions commonly used in characterization of extreme events in nonlinear processes in geosciences.