# How to estimate this expectation involving two binomial random variables?

$$X$$ 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.