Discrete analogues of continuous bivariate probability distributions

In many real-world applications, the random variables modeling the phenomena of interest are continuous in nature, but their observed values are actually discrete and hence it is reasonable and convenient to choose an appropriate multivariate discrete distribution generated from the underlying continuous model preserving one or more important features. In this work, two methods are discussed for deriving a bivariate discrete probability distribution from a continuous one by retaining some specific features of the original stochastic model, namely 1) the joint density function, or 2) the joint survival function. These methods can be regarded as the bivariate extension of two popular methods used for deriving a univariate discrete distribution from a continuous one; they can be also used as viable alternatives to extant techniques of construction of bivariate discrete random variables. Examples of applications are presented, which involve two types of bivariate exponential distributions and a bivariate Pareto distribution, in order to illustrate how the procedures work and show that some bivariate discrete distributions that were recently proposed in the literature can be actually regarded as discrete counterparts of well-known continuous models. A numerical study is presented in order to illustrate how the procedures are practically implemented and to present inferential aspects. A real dataset is eventually fitted using two discrete analogues of a bivariate exponential distribution.