A Note on Positive Dependence and the Structure of Bivariate Distributions

Abstract

Let X and Y be random variables whose marginal distributions $\mu ( \cdot )$ are identical. If their joint distribution function admits a series expansion in appropriate sets of functions with nonnegative coefficients, it is shown that \[ P\{ X \in A,Y \in A\} \geqq P\{ X \in A\} P\{ Y \in A\} \] for every $\mu $-measurable set A. This property is demonstrated for a variety of known two-dimensional distributions whose marginal measures are either discrete or absolutely continuous with respect to Lebesgue measure. Examples include bivariate Gaussian, exponential, gamma, chi-square, binomial, Poisson, negative binomial, and hypergeometric distributions.