Bounds for covariances and variances of truncated random variables

Abstract

We show that a lower bound for covariance of $\min (X_1,X_2)$ and $\max (X_1,X_2)$ is $\cov{X_1}{X_2}$ and an upper bound for variance of \\ $\min (X_2,\max (X,X_1))$ is $\var{X} + \var{X_1} +\var{X_2}$ generalizing previous results. We also characterize the cases where these bounds are sharp.

7 pages. Revised during October 2002