Let $X$ be a random variable taking values on the real line.
Let $R(X) = max\{0, X\}$. Is it true that the covariance $Cov[X, R(X)] \ge 0$ irrespective of the distribution of $X$? Many experiments, as well as the intuition seem to suggest that their covariance must be non-negative. Is it true? If so, how can I prove it?



On estimating Covariance between a random variable and its non-linear transform
$${\rm Cov}\,(X,R(X))= \int\_0^\infty x^2P(x)\,dx - \left(\int\_{-\infty}^\infty xP(x)\,dx\right)\left(\int\_{0}^\infty xP(x)\,dx\right)$$
$$=\int\_0^\infty x^2P(x)\,dx - \left(\int\_{0}^\infty xP(x)\,dx\right)^2- \left(\int\_{-\infty}^0 xP(x)\,dx\right)\left(\int\_{0}^\infty xP(x)\,dx\right)$$
$$={\rm Var}\,R(X)- \left(\int\_{-\infty}^0 xP(x)\,dx\right)\left(\int\_{0}^\infty xP(x)\,dx\right)\geq 0.$$
The variance is positive by definition and the second term is minus the product of a negative and a positive integral.


