X and Y are two independent binomial random variables where X~ B(K_1, q), Y~ B(K_0, p), (Note: p+q is not necessarily equal to 1). I am wondering how to compute the following expectation (the expectation is taken over random variable X): E[X Pr(Y >= Xq/p)] = \sum_{t=0}^{K_1} t Pr(Y >= tq/p) Pr(X=t). Or at least give an upper and lower bound.
How to estimate this expectation involving two binomial random variables?
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