The Bivariate Poisson–X–Exponential Distribution: Theory, Inference, and Multidomain Applications

We propose the Bivariate Poisson–X–Exponential Distribution (BPXED), a flexible bivariate count model obtained by compounding Poisson variables with a shared X–Exponential latent mixing distribution. The model extends the Poisson–X–Exponential (PXED) distribution and includes several bivariate Poisson-type models as special or limiting cases. Closed-form expressions are derived for the joint probability mass function, probability generating function, moments, and covariance structure, showing that dependence arises from shared latent heterogeneity and is restricted to positive correlation. Parameter estimation is developed using maximum likelihood, regression-based, and Bayesian approaches, and a Monte Carlo simulation study demonstrates a good finite-sample performance. Applications to soccer scores, reliability failures, and correlated photon counts illustrate improved goodness-of-fit over classical and recent competing models. Overall, BPXED provides an analytically tractable and interpretable framework for modeling positively dependent and overdispersed bivariate count data.