Abstract
A non-linear first-order differential equation is proposed 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, 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.
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Cheng, Q. (2016). A First-order Non-linear Differential Equation Characterizing Multiple Types of Probability Distributions. In: Raju, N. (eds) Geostatistical and Geospatial Approaches for the Characterization of Natural Resources in the Environment. Springer, Cham. https://doi.org/10.1007/978-3-319-18663-4_148
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DOI: https://doi.org/10.1007/978-3-319-18663-4_148
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18662-7
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