Bivariate Distributions on Z2

This study introduces three new bivariate distributions on Z 2 with a discussion of the main properties of these distributions. The correlation in these distributions varies in range; some have a full range while others can take only positive or negative values. The development of these distributions depends on a trivariate reduction and the extension of the Moran results. Two real applications—the difference in the number of casualties to the number of employees on duty on railroads and the difference in the number of goals scored in the English Premier League in different years—are fitted to the proposed bivariate distributions.