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Simplify the following expression

$$ \iint_{-\infty}^{c+x}xf(x)f(y) \,dy\,dx+\iint_{c+x}^{\infty}yf(x)f(y) \,dy\,dx $$

where $x$ and $y$ are iid random variables; $c$ is a constant; and $f$ is the probability density function. You may call the CDF as $F(\cdot)$.

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up vote 1 down vote accepted

This is $\mathbb E(Z)$, where $$ Z=X\cdot\mathbf 1_{Y\leqslant c+X}+Y\cdot\mathbf 1_{Y\geqslant c+X}. $$ Since $X$ and $Y$ are identically distributed, this is also $\mathbb E(T)$, where $$ T=X\cdot(\mathbf 1_{Y\leqslant X+c}+\mathbf 1_{Y\leqslant X-c}). $$ And finally, this is also $$ \mathbb E(\max\{X,Y\})-\mathbb E(Y-X\,;\,0\leqslant Y-X\leqslant c). $$

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