In the work, we propose and study a new two-parameter compound G family of continuous distributions. Relevant statistical properties are mathematically derived. Many new G families of bivariate distributions are presented using Renyi's copula, Clayton copula, Ali-Mikhail-Haq copula, Farlie-Gumbel-Morgenstern copula, and modified Farlie-Gumbel-Morgenstern copula. Based on a special case we presented a new Lomax extension and studied its relevant statistical properties. The maximum likelihood method is used and employed for estimating the model parameters. Two applications to real-life data sets are presented for illustrating the superiority of the new family.