I have two random variables $X$ and $Y$, both receiving values between 0 and 1.
I know that $E[X - Y] \ge 0$.
Can I get any inequality of the form:
$P(X - Y \ge \delta) \le F(\delta,X,Y)$
where $F(\delta,X,Y)$ is a (reasonable) function of $\delta$ and $E[X-Y]$?
Markov inequality would be good here, for example, by setting $F(\delta,X,Y) = E[X-Y]/\delta$. However, Markov inequality would require $X-Y \ge 0$, and I do not necessarily have that.