Extended Gamma, Beta and Hypergeometric Functions: Properties and Applications

Extended gamma, beta and hypergeometric functions

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

The main objective of this paper is to study the newly introduced extended gamma function and present some properties of the existing extended beta and hypergeometric functions with their properties such as integral formulas, functional relations, differential formulas, summation formulas, recurrence relations and the Mellin transform. In addition, applications of the extended beta function to statistical distributions have been discussed by providing mean, variance, coefficient of variance, moment generating function, cumulative distribution and reliability function of the new extended beta distribution.