On the Covariance between Functions

The covariance between the functions of two random variables is obtained in terms of the cumulative distribution function. This result generalizes previous formulae given by W. Hoeffding (1940, Schriften Math. Inst. Univ. Berlin5, 181–233) and K. V. Mardia (1967, Biometrika54, 235–249). An expansion for the covariance, an inequality, a maximum correlation and other consequences are obtained from this generalization.