In this paper, we introduce two families of general bivariate distributions. We refer to
these families as general bivariate exponential family and general bivariate inverse
exponential family. Many bivariate distributions in the literature are members of the
proposed families. Some properties of the proposed families are discussed, as well as a
characterization associated with the stress-strength reliability parameter, R, is presented.
Concerning R, the maximum likelihood estimators and a simple estimator with an explicit
form depending on some marginal distributions are obtained in case of complete
sampling. When the stress is censored at the strength, an explicit estimator of R is also
obtained. The results obtained can be applied to a variety of bivariate distributions in the
literature. A numerical illustration is applied on some well-known distributions. Finally a
real data example is presented to fit one of the proposed models.