Suppose that $X \sim \textrm{Bin}(n,p)$ and $Y \sim \textrm{Bin}(n,p)$ are independent and identically distributed variables.
Is there a nice way to calculate the expectation of their absolute difference, i.e., $\mathbb{E}\left|X - Y\right|$? or even a good upper bound on it?
From some searching, (e.g., link) it looks like the overall distribution has no simple form but I'm wondering if it might be easier to find the expected value.
Another potentially relevant work I found is https://www.cs.bgu.ac.il/~karyeh/sharp-mad.pdf, which calculates bound on a closely related value $\mathbb{E}\left|X - \mathbb{E}(X)\right|$. For $p$ which is not too close the endpoints, $\mathbb{E}\left|X - \mathbb{E}(X)\right|$ is bounded by the s.d. of $X$.
I'm wondering if some similar $\Theta(\sqrt{n})$ bound would also work for $\mathbb{E}\left|X - Y\right|$.
A generalization where $X \sim \textrm{Bin}(n_1,p_1)$ and $Y \sim \textrm{Bin}(n_2,p_2)$ are still independent would also be useful.