$X$ and $Y$ are two independent binomial random variables where $X\sim B(K_1, q), Y\sim B(K_0, p)$, (Suppose $p>q, p+q$ is not necessarily equal to 1). I am wondering how to compute the following expectation: $$ \mathbb{E}_X[X \ \text{Pr}(Y\geq \frac{q}{p}X)] = \sum_{t=0}^{K_1} t \text{Pr}(Y\geq \frac{q}{p}t) \text{Pr}(X=t).$$ Or at least give an upper and lower bound.

I am also curious in another problem $\mathbb{E}_X[X \ \text{Pr}(Y\geq \frac{p}{q}X)]$. If we assume that $K_0=K_1$, then $Y$ and $\frac{p}{q}X$ have the same mean. This might be a more interesting case.


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