<p>To handle data that involve counting elements with variable characteristics, the paper develops a first-order Poisson random coefficient mixed thinning integer-valued autoregressive time series model. Moments and autocovariance functions for this model are studied since the distribution of the innovation sequence is Poisson. Based on these theoretical foundations, the conditional least squares and conditional maximum likelihood methods are employed to estimate the parameters of the model. Asymptotic properties of the obtained estimators are established. The performances of these estimators are evaluated via simulations, and the conclusion is drawn that the conditional maximum likelihood method is superior. Furthermore, a set of crime data from the town of Inverell in New South Wales, Australia, is analyzed to demonstrate the practical relevance of the model.</p>