DEPENDENCE AND UNCORRELATEDNESS

In this note we prove the following. Theorem.Let F and G be two one-dimensional distributions with nonzero and finite variances.There exist two random variables X and Y such that 1) X has distribution F and Y has distribution G,2) X and Y are dependent,3)the correlation coefficient of X and Y is zero if and only if at least one of F and G is not a two-point distribution.