Ordinary Differential Equations with Their Solutions as Probability Functions of Extended Rayleigh Distributions

Abstract

In this paper, ordinary differential equations (ODEs), whose solutions are the probability functions such as cumulative distribution function (CDF), probability density function (PDF), quantile function (QF), survival function (SF), hazard function (HF), and reversed hazard function (RHF) of extended  Rayleigh distributions, that are obtained by applying differential calculus and quantile mechanics along with sufficient algebraic interpretations. New extended Rayleigh distribution to higher dimensions, extended Rayleigh-Rayleigh distribution, and Power Rayleigh distribution are of interest. The necessary conditions required for the existence of the ODEs and the initial (or boundary) conditions consistent with the various parameters that defined the distributions are also included. Furthermore, the particular ODEs solved by each probability function of the selected distributions are derived and displayed for specific parameter values. The method employed can serve as a new technique for approximating and estimating solutions of ordinary differential equations, which can be used in modelling various dynamical systems that are difficult to solve with many available methods by manipulating the rich body of knowledge in probability distribution theory as a valuable tool, thereby strengthening the interdisciplinary connection between mathematics and probability theory, statistics.