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Think of two iid random variables $x$ and $y$ with density $f$ and CDF $F$ and a constant $c$. What could the qualitative meaning of the following expression be?

$$\iint_{-\infty}^{c+x}xf(x)f(y)dydx=\int x\left[\int_{-\infty}^{c+x}f(y)dy\right]f(x)dx=\int xF(c+x)f(x)dx$$

Any suggestion, including how to simplify this expression, will be appreciated!!

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    You might consider how you might calculate the expected value of $X$ given that $Y-X = c$2012-10-31
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    @Henry Can you please offer a little more hint? Thanks2012-10-31
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    Think that $F(c+x)=P(Y \leq c+X)$2012-10-31
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    What if we rather have $$\iint_{-\infty}^{c+x}f(x)f(y)dydx$$2012-11-05

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Let $A=[Y\lt X+c]$, then the integral is $\mathbb E(X\mathbf 1_A)$.