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!
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