Alec's expected value trick

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Lemma

Suppose is a non-negative real random-variable, then I have found a way to calculate the expected value of $X$, $\E{X}$:

• We claim: $$\E{X}\eq\int^\infty_{0}\big(1-\P{X<x}\big)\mathrm{d}x$$ or $$\E{X}\eq\int^\infty_0\P{X\ge x}\mathrm{d}x$$

Statement

We should be able to use this for a general real random variable by considering conditional random variables (via conditional probability) for the positive and negative parts respectively.