Let $X$ and $Y$ are two independent binomial random variables where $X\sim B(K, q), Y\sim B(K, p)$. I am wondering how to compute or estimate the following expectations: $$ \ \mathbb{E}[|pX-qY|]\ \ \text{and}\ \ \mathbb{E}[\text{ReLU}(pX-qY)].$$
where ReLU($x$) = max($x, 0$) Furthermore, what does the distribution of $pX-qY$ look like?
Update: Actually I'm more interested in $\mathbb{E}[pX\mathbb{1}(pX-qY>0)]\ \text{and}\ \mathbb{E}[qY\mathbb{1}(pX-qY>0)]$, where $\mathbb{1}$ is the indicator function. We know that $\mathbb{E}[pX\mathbb{1}(pX-qY>0)]-\mathbb{E}[qY\mathbb{1}(pX-qY>0)]=\mathbb{E}[\text{ReLU}(pX-qY)]$.