Monday, July 4, 2016

calculus - Understanding mathbbE((Xc)mathbb1X>c)=int+inftycmathrmP(Xgeqslantt),mathrmdt

For example, in this MSE post, it is claimed that the following holds:



E(|Xc|)=cP(X



The modulus part is clear, so it suffices to stick with the first part, namely:




\mathbb{E}((X-c) \mathbb{1}_{\{X > c\}}) = \int_c^{+\infty}\mathrm P(X\geqslant t)\,\mathrm dt



Why does this hold? What I'm seeing is that:
\mathbb{E}((X-c) \mathbb{1}_{\{X > c\}}) = \int_{c}^{\infty}(u - c) \mathrm dF(u) = \int_{c}^{\infty} \int_{0}^{u-c} 1 \mathrm dw \mathrm dF(u) = \int_{c}^{\infty} \left(\int_{0}^{u-c} \mathrm dF(u) \right) \mathrm dw



but can't make the last connection, especially since in my case integral limit depends on integrand value (using same u), hence I'm not even sure that such switching of integral order is possible.



What am I missing?

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