Intuition behind expected value formula? [duplicate]

I just learned the expected value formula for a positive random variable: $$E[X] = \int_0^\infty P(X>x)dx$$.

I can follow the proof but I'm a little unclear as to what the intuition behind this formula is. Can someone formulate it in a more intuitive manner?

marked as duplicate by Lee David Chung Lin, Mike Earnest, Brahadeesh, Cesareo, José Carlos SantosApr 17 at 12:51

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• See what happens in case of a random variable taking positive integer values. – Ignorant Mathematician Apr 15 at 16:02
• You are integrating the probability of $X$ being large. The more likely $X$ is to be large, the larger it is on average. – Mike Earnest Apr 15 at 16:02