Discrete analogues of bivariate continuous distributions

Modeling correlated count data through some bivariate (or multivariate) discrete distribution is essential in many real-world applications in a wide range of fields, such as industrial quality control, healthcare, marketing, management science, and many others. Moreover, finding a discrete analogue to a continuous distribution can be useful in those problems where dealing with a continuous model is computationally cumbersome and substituting it with an appropriate discrete model can produce an approximate but still acceptable solution with a much smaller computational effort. In this work, two methods for deriving a bivariate discrete distribution from a bivariate continuous distribution are discussed, which retain the expression of either the joint density function or the joint survival function. These methods can be regarded as the bivariate extension of two popular methods used for deriving a discrete counterpart of a continuous distribution; they can be used as viable alternatives to existent techniques of construction of bivariate discrete random variables. Examples of application to several continuous distributions are presented in order to illustrate how the procedures work. A real dataset taken from the literature is eventually analyzed and fitted using two discrete analogues of a bivariate exponential distribution