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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?

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    Are you sure about your finding? For example, if $f=g$, then $D=1$ and $P(Y\leqslant X)=\frac12$.2012-07-05
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    Where does the factor of $2$ come from in your expression for Pr($Y \le X$)?2012-07-05
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    Sorry, I now fixed the factor of 2.2012-07-05
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    It's not a duplicate since now I am asking about showing the approximation formally, and quantifying it...2012-07-05
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    @Dilip Thanks for the information. Asking again and again small variations of the same question seems to be all the rage these days... (About the condition $E(Y)>E(X)$, you are right but one can modify slightly $f$ to get some $g$ with $E(Y)>E(X)$, $P(Y\ge X)$ close to $\frac12$ and the former $D$ close to $1$.)2012-07-05

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