Sharp lower and upper bounds for the covariance of bounded random variables

In this paper we derive sharp lower and upper bounds for the covariance of two bounded random variables which are applicable when knowledge about their expected values, variances or both is available. When only the expected values are known, our result can be viewed as an extension of the Bhatia–Davis inequality for variances. We also provide a number of different ways to standardize covariance. For a binary pair of random variables, one of these standardized measures of covariation agrees with a frequently used measure of dependence between genetic variants.