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**slevvio** Let X be a discrete random variable that can only take non-negative integers (i.e. $\displaystyle R_x \subset \mathbb{N} $).

Show that $\displaystyle \mathbb{E}(X) = \sum_{x=0}^{\infty}\mathbb{P}\{X>x\}$.

I don't know how to work this out and I would appreciate any help. I know that $\displaystyle \sum_{x=0}^{\infty}\mathbb{P}\{X>x\} = \sum_{x=0}^{\infty} 1 - F(x) $ where F(x) is the CDF but I cannot see how even that gives the expectation. Any help would be appreciated and thanks in advance.