Saturday, February 11, 2017

probability - Expected value of the negative part of a random variable

I want to prove that, if X is a real valued random variable with finite expected value, then:



E[X]=0P(Xt)dt0P(Xt)dt.



We have that E[X]=E(X+)E(X)
and I know how to prove that if Y is a non-negative r.v., then its expected value can be expressed as E[Y]=0P(Yt)dt.



I am having trouble expressing the second integral as the expectation of the negative part, X.




Can anyone help me with that?

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