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 Santos Apr 17 at 12:51

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