<p>Count data modeling is a critical aspect across various disciplines. However, traditional models, such as first-order integer-valued autoregressive processes, often struggle to capture the inherent variability in real-world scenarios. The incorporation of a randomized thinning operator in the first-order integer-valued autoregressive process addresses these limitations. This paper introduces a novel stationary first-order integer-valued autoregressive process with a random coefficient having uniform Poisson–Ailamujia distributed marginals. This process encompasses the first-order integer-valued autoregressive process with binomial thinning as a particular case. The paper presents a comprehensive exploration of the statistical properties of the process. Furthermore, we explore distinct parameter estimation approaches and forecasting methods, enhancing the modeling capabilities of the process, with performance evaluation conducted through a simulation study. The study concludes with a comparative analysis and practical application of the proposed model to real-world data, validating its adaptability and potential across diverse applications in the context of the first-order integer-valued autoregressive process with a random coefficient.</p>