I have a question related to distance between two CDF's.
Suppose we have CDF $F$ and $G$: An RV $X$ follows $F$ and $Y$ follows $G$, with the same support, say $[0,1]$.
If we match each $x$ to $G^{-1}(F(x))$ (matching points that have the same quantile), what would be the measure of $x$'s whose distance to its match is closer than the average distance?
That is,
$$\mathbf{Prob}[x|(x-G^{-1}(F(x)))^2<\mathbb{E}(x-Y)^2].$$
Will there be any simple expression for this probability?