Let $X,Y \sim \text{Bin}(n,0.5)$ for some positive $n$.
What is a lower bound for $\mathbb{E}(XY)$? When is it achieved?
My try:
I got confused by the following two results and couldn't proceed!
By Jensen's, $\mathbb{E}(XY) \geq \mathbb{E}(X)\mathbb{E}(Y)$. But we also know that $\text{cov}(X,Y)=\mathbb{E}(XY) - \mathbb{E}(X)\mathbb{E}(Y)$, and this quantity can be negative!
Please help me to proceed. Thanks in advance!
Hint: If $\rho_{X,Y}:=\frac{\text{cov}(X,Y)}{\sigma_X\sigma_Y}$ denotes the correlation coefficient of $X,Y$, then you know that $$-1\le \rho\le 1$$ Now $\sigma_X=\sigma_Y=0.5\sqrt{n}$ and $\mathbb E[X]=\mathbb E[Y]=0.5n$. Hence, $$-1\le \rho=\frac{\mathbb E[XY]-(0.5n)^2}{\left(0.5\sqrt{n}\right)^2} \implies 0.25n(n-1)\le \mathbb E[XY]$$ The equality is attained when $\rho=-1$, for instance when $Y=n-X$ (negative linear relationship with "slope" -1).