Discrete analogs of a bivariate Pareto distribution

In many real-world problems, the phenomena of interest are continuous in nature and modeled through well-established continuous probability distributions, but it often occurs that observed values are actually discrete and then it would be more appropriate to use a (multivariate) discrete distribution generated from the underlying continuous model, which preserves one or more of its important features. In this work, we illustrate the genesis and properties of two bivariate discrete distributions that can be derived as discrete counterparts to a bivariate continuous Pareto distribution. The two discrete probability distributions preserve the expression of either 1) the joint density function or 2) the joint survival function of the parent distribution at each pair of non-negative integers. Their joint and marginal probability mass functions are derived and compared; the expressions for their bivariate failure rate vectors are also obtained. This study reveals how the second discrete analog is easier to handle with respect to the first one, whose expression of the probability mass function involves the Riemann zeta function.