Classes of Ordinary Differential Equations Obtained for the Probability Functions of Gompertz and Gamma Gompertz Distributions

Abstract

In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODE) for the probability density function, quantile function, survival function, inverse survival function, hazard function and reversed hazard function of the Gompertz and gamma Gompertz distributions. The stated necessary conditions required for the existence of the ODEs are consistent with the various parameters that defined the distributions. Solutions of these ODEs by using numerous available methods are a new ways of understanding the nature of the probability functions that characterize the distributions. The method can be extended to other probability distributions and can serve an alternative to approximation especially the cases of the quantile and inverse survival functions. Finally, the link between distributions extended to their differential equations as seen in the case of the ODE of the hazard function of the gamma Gompertz and exponential distributions