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This paper proposes a new family of bivariate distributions designed to jointly model one continuous and one discrete random variable using the Farlie-Gumbel-Morgenstern (FGM) copula. Such mixed distributions are increasingly important in real-world applications where time-based and count-based data types interact in domains like telecommunications, technical maintenance, and customer services instances. The proposed framework represents a flexible structure for constructing a wide class of joint distributions by specifying the continuous and discrete variables independently. This paper starts with the presentation of a general formula and its properties including joint, marginal, conditional, and hazard functions are thoroughly derived. The Bivariate Exponential–Geometric (BEG) distribution is scrutinized as a special case, and several theoretical features such as the moment-generating function and correlation behavior are explored. Estimation techniques including Maximum Likelihood Estimation (MLE), the Method of Moments (MoM), and Bayesian estimation are proposed. A simulation study is conducted with the purpose of evaluating the performance of each method. The conclusive remarks of the study demonstrate that MLE consistently yields efficient estimates, while Bayesian estimation performs well under informative priors. A real-world application involving banking data also proves the practical relevance and applicability of the proposed model. This new bivariate distribution family constitutes both theoretically sounding and practically efficient tools for modeling mixed data types. It can be also further extended to be applied to multivariate settings involving both continuous and discrete components. Beyond theoretical contributions, the framework also offers practical value by supporting evidence-based decision-making in fields such as environmental monitoring, healthcare, and socio-economic planning, thereby strengthening its relevance to real-world development challenges.