most recent 30 from http://mathoverflow.net 2013-05-24T13:52:15Z http://mathoverflow.net/feeds/user/27084 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109106/upper-bound-on-expectation-value-of-the-product-of-two-random-variables/109142#109142 Arturo Erdely 2012-10-08T12:41:08Z 2012-10-08T12:50:44Z

<p>I would try yo apply Hoeffding's Lemma, who used his result to identify the bivariate cdfs with given marginal cdfs that minimize or maximize correlation. Let $(X,Y)$ be a random vector with bivariate cdf $H$, let $F$ and $G$ be their marginal cdfs, respectively. It is well known that a sharp upper bound for $H(x,y)$ is $\min(F(x),G(y))$. By Hoeffding's Lemma we get that $$E(XY)\leq E(X)E(Y)+\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\big[\min(F(x),G(y))-F(x)G(y)\big]dxdy$$</p>