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A form of cumulative distribution
Let $X$ and $Y$ be two continuous independent RVs with $f(x)$ and $g(x)$ as probability density functions, respectively. Assume that $E[Y]>E[X]$. Now, I have found numerically that the expression: $D=\frac{1}{2}\int_{-\infty}^{\infty} \min(f(x),g(x)) dx $ which describes half the 'overlap area' of the two densities, is a rough approximation of: $ \Pr (Y \le X) = \int _{-\infty}^{\infty}\left[f(x)\int _{-\infty}^{x}g(y)\,dy\right]\,dx $
How can I formally show that the approximation holds for any two densities for which $E[Y]>E[X]$, and how can I quantify the strength of the approximation?