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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 $Y-X = c$ – Henry Oct 31 '12 at 9:20
@Henry Can you please offer a little more hint? Thanks – Daniel Lårs Oct 31 '12 at 9:29
Think that $F(c+x)=P(Y \leq c+X)$ – Alex Oct 31 '12 at 11:42
What if we rather have $$\iint_{-\infty}^{c+x}f(x)f(y)dydx$$ – Daniel Lårs Nov 5 '12 at 19:26
up vote 0 down vote accepted

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

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