@JamesMartin I fixed the problem statement. Does it make more sense now?

Thanks for the comment! That's a very important part of the question that I accidentally left out in the revision. All elements are hashed using the same hash function and M_i is the X smallest elements of L_i by hash value. I've updated the question with this info. Notably this means that C and M_i are correlated.

@MattF. No this doesn't miss anything. I would just add that we want to minimize the value of $X$. Feel free to update the question with this formulation. Otherwise I will update it tomorrow.

Thanks, I edited the problem statement such that $c$ is gone and $\epsilon$ is some error bound that you can determine (instead of a parameter that is an input to the problem).

ah yes, I'd forgotten that the variables are continuous, not discrete, so the event $X = Y$ has zero-probability of happening. If all the variables are discrete, then we can show $P(X + Z \leq x, X + A \leq x) \geq P(X + Z \leq x, Y + A \leq x)$ by using the fact that $P(X + Z \leq x, X + A \leq x) = P(X + Z \leq x, Y + A \leq x | X = Y)$.

Thanks for the response, it's very nice! I have one more question. Would it be correct to say $\Pr[(X \leq x) \cap (Z \leq x - X) \cap (A \leq x - X)] = \Pr[(X \leq x) \cap (Y \leq x) \cap (Z \leq x - X) \cap (A \leq x - Y) | X = Y]$ and the inequality follows directly from that?