Some Problems Connected with the Distributions of Products and Ratios

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.

In this paper, the distributions of products and ratios involving real scalar positive variables, symmetric products and symmetric ratios of real positive definite matrices and symmetric product and symmetric ratios of Hermitian positive definite matrices in the complex domain are considered. Real scalar variable case is taken as a starting point. Here, it is pointed out how distributions of products and ratios are connected to Bessel integrals, reaction-rate probability integrals in nuclear reaction-rate theory, inverse Gaussian density, Kr¨atzel integral and transform, pathway transform, fractional integrals of the first and second kind etc. Only one
illustrative example each is given. Then, symmetric product and symmetric ratios of positive definite matrices in the real and complex domain are considered. Here also one illustrative example each is given. Some current research materials are also
mentioned in the introduction.