Discretization of Random Variables with Applications

Abstract

Mostly discrete distributions have been derived and studied independently of any continuos counter part. But sometimes we find that some properties of a continuous distribution, if applied in the discrete support, also characterises a discrete distribution. In these situations we call the discrete distribution as the discretized version of the continuous distribution. For example, Geometric distribution is the discrete version of the Exponential distribution. Many authors have discretized continuous random variables by using different properties of the continuous distributions viz. distribution function, moments, characterizing properties etc. They have also shown various applications of these discretized random variables. For example, in a complex system, the system response function is a random variable due to random nature of the component lives. But for the complexity of the system, generally analytical expressions for the response function are difficult to obtain. In such situations discretization of the continuous variables helps us to study the same. Here we have discretized the Weibull distribution by using its failure rate function. Application of the discretization has been demonstrated in approximating the reliability of Solid-shaft, a well-known engineering item. Numerical study shows that the proposed discretization gives a better approximation of the system reliability over the methods of discrete concentration, which is based on survival function, moment equalization and frequency curve equalization.

The discrete new XLindley distribution and the associated autoregressive process

“…For example, the Poisson distribution models count data showing equidispersion but count data frequently also exhibit overdispersion or underdispersion, which has led to the development of more Ćexible models during the last decades. In this regard, different methods of generating discrete probability distributions as analogues of continuous probability distributions have been introduced in the statistical literature, such as the inĄnite series discretization method (Good (1953), Kulasekera and Tonkyn (1992), Sato et al (1999)), the survival discretization approach (Nakagawa and Osaki (1975)), the reversed hazard function discretization method (Ghosh et al (2013)) and the compound two-phase method (Chakraborty (2015)), among others. The reader is referred to Chakraborty (2015) for a survey on discretization methods of continuous distributions, where their main differences and implications can be found.…”

Modelling Count Variables: A Comparative Analysis of two Discretization Techniques

“…The process involves using different mathematical concepts to derive discrete analogous to continuous distributions. Different approaches to discretizing a continuous distribution have been developed [2][3][4]. Among the prominent techniques for achieving this is the survival function of the continuous distribution, as was first used on the Weibull distribution [5][6].…”