The bivariate probability distributions with periodic failure rate via hyperbolic differential equation

The bivariate random vectors with almost lack of memory property is a new area of investigation in applied studies. It has been shown that they possess a periodic (with respect to both arguments) failure rate, and have a specific form for its probability distribution. We discuss the question how complete a periodic bivariate failure rate function will characterize this class of probability distributions. It is shown that bivariate failure rates (positive functions with integrable square on the first quadrant) and marginal distributions along the coordinate axes uniquely determine a two-dimensional random vector under some minimal restrictions. More details are given for the case of periodic bivariate failure rates. It is proven that the latter characterize bivariate distributions with almost lack of memory property. As a special case, the bivariate exponential distribution with constant failure rate is discussed. Some possible applications are briefly comment. components is derived.