Ordinary differential equations of probability functions of convoluted distributions
No Thumbnail Available
Date
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
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.
Keywords
QA Mathematics