Friday, August 30, 2019

probability - Expectation of nonnegative Random Variable





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Can someone help me give me some pointers as to how to prove this relation?


Answer



Let p be the probability measure. We have that $\int_{0}^{\infty}\left[1-F\left(x\right)\right]dx=\int_{0}^{\infty}\Pr\left[X>x\right]dx=\int_{0}^{\infty}\left[\int1_{X>x}dp\right]dx $ using Fubini's theorem we have $\int_{0}^{\infty}\left[\int1_{X>x}dp\right]dx=\int\left[\int_{0}^{\infty}1_{X>x}dx\right]dp=\int Xdp=E\left[X\right] $


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