I want to prove that, if X is a real valued random variable with finite expected value, then:
E[X]=∫∞0P(X≥t)dt−∫0−∞P(X≤t)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(Y≥t)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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