Developing bivariate distribution models for situations involving discrete and continuous random variables

This paper proposes a novel and flexible framework for constructing bivariate distribution models in the mixed setting where one variable is discrete and the other is continuous. The proposed approach enables the systematic development of analytically tractable joint distributions with explicit marginals and conditional structures. To illustrate its applicability, two bivariate models are developed and fitted to a real-world dataset on average credit card expenditure and number of derogatory reports. Explicit closed-form expressions for the joint moments are derived, and the dependence structure is rigorously examined through conditional behavior, local dependence properties, and stochastic ordering. Furthermore, important distributional properties of the concomitants of order statistics arising from the proposed models are established.