On a Bivariate Lack of Memory Property Under Binary Associative Operation

Abstract

A binary operation * over real numbers is said to be associative if (x*y)*z = x*(y*z) and it is said to be reducible if x*y = x*z or y*w = z*w if and only if z = y. The operation * is said to have an identity element ẽ if x*ẽ = x. Roy [Roy, D. (2002). On bivariate lack of memory property and a new definition. Ann. Inst. Statist. Math. 54:404–410] introduced a new definition for bivariate lack of memory property and characterized the bivariate exponential distribution introduced by Gumbel [Gumbel, E. (1960). Bivariate exponential distributions. J. Am. Statist. Assoc. 55:698–707] under the condition that each of the conditional distributions should have the univariate lack of memory property. We generalize this definition and characterize different classes of bivariate probability distributions under binary associative operations between random variables.