A Characterization of Joint Distribution of Two-Valued Random Variables and Its Applications

We obtain an explicit representation for joint distribution of two-valued random variables with given marginals and for a copula corresponding to such random variables. The results are applied to prove a characterization of r-independent two-valued random variables in terms of their mixed first moments. The characterization is used to obtain an exact estimate for the number of almost independent random variables that can be defined on a discrete probability space and necessary conditions for a sequence of r-independent random variables to be stationary.