Suppose that I have two non-negative real valued random variables $x, y \in Z\_+$ that always satisfy $$x+y \leq 1.$$ Also suppose that $E[x] = 1/2$ and $E[y] = 1/4$. What is the maximum possible value of $E[xy]$? Can it be larger than $1/8$?


More generally, is there a systematic way of analyzing $E[xy]$ for say other values of $E[x]$ and $E[y]$? (You can assume the assumptions $x, y \in Z\_+$ and $x+y\leq 1$ continue to hold.)



Expectation of multiplied random variables given their individual expectations
It is possible to have $E[xy]=9/64$, by
$$P\left[(x,y)=\left(\frac38,\frac18\right)\right]=\frac12$$
$$P\left[(x,y)=\left(\frac58,\frac38\right)\right]=\frac12$$


This can be guessed by knowing that optimal probability distributions are often normal, uniform, or concentrated at two points.


I don't have a proof that this is optimal, but here is a useful lemma: If $(x,y)$ and $(x',y')$ both have density at least $\epsilon$ in an optimal set-up, then the combination of $x<x'$ and $y>y'$ is impossible. The proof is that if those two inequalities both hold, we could increase $E[xy]$ by placing the density on $(x,y')$ and $(x',y)$ instead.


To actually prove that the above two-point solution is optimal, you'd probably apply arguments like the lemma to show that any optimal distribution must be concentrated on some curve with non-decreasing $x$ and non-decreasing $y$; then that it must be concentrated on some line segment; and then that it must be concentrated on two points.


