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# Ordinary differential equations of probability functions of convoluted distributions

Okagbue, H. I. and Adamu, M. O. and Anake, T. A. (2018) Ordinary differential equations of probability functions of convoluted distributions. International Journal of Advanced and Applied Sciences, 5 (10). pp. 45-52.

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## Abstract

Convolution is the sum of independent and identically distributed random variables. Derivatives of the probability density function (PDF) of probability distribution often lead to the construction of ordinary differential equation whose solution is the PDF of the given distribution. Little have been done to extend the construction of the ODE to the PDF, quantile function (QF), survival function (SF), hazard function (HF) and the reversed hazard function (RHF) of convoluted probability distributions. In this paper, three probability distributions were considered namely: Constant parameter convoluted exponential distribution (CPCED), convoluted uniform exponential distribution (CUED) and different parameter convoluted exponential distribution (DPCED). First order ordinary differential equations whose solutions were the PDF, SF, HF and RHF for the probability functions of CPCED by the use of differential calculus. The case of the QF was second order nonlinear differential equations obtained by the use of Quantile Mechanics. Similarly, the same was obtained for CUED for the two cases of the distribution. Some new relationships were obtained for the PDF, SF and HF, and also the RHF, PDF and CDF with their corresponding first derivatives. The difficulty of obtaining the ODE for the probability functions of the DPCED was due to the different parameters that characterize the distribution. The use of partial different equations is not an alternate because the distribution has only one independent variable.

Item Type: Article Convolution Parameter Exponential Uniform Calculus Quantile mechanics Q Science > QA Mathematics Faculty of Engineering, Science and Mathematics > School of Mathematics Mrs Patricia Nwokealisi 27 Aug 2019 10:37 27 Aug 2019 10:37 http://eprints.covenantuniversity.edu.ng/id/eprint/12844 View Item