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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?

Greenteamaniac
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  • $y$ denotes the average distance? And distance is defined as square or absolute value? (not sure if this makes any difference though) – Jimmy R. Nov 07 '18 at 06:09
  • $\mathbb{E}[(x-y)^2]$ is the average distance to $Y$ for a given realization of $x$. And I just use the square distance. – Greenteamaniac Nov 07 '18 at 06:17

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