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I once read the following inequality

enter image description here

Is there any specific name for this inequality? And, how to prove it?

user3125
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    That's *NOT* an inequality, it's an equality. There's a clear "$=$" sign between $\mathbb{E}X$ and the $\int$. – Glen_b Oct 18 '13 at 19:43
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    Use Fubini. Then, pause. Then, ask yourself why the argument doesn't work, in general, if $X$ can take both positive and negative values. – cardinal Oct 18 '13 at 20:01
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    See discussion of the result [here](http://stats.stackexchange.com/questions/18438/does-a-univariate-random-variables-mean-always-equal-the-integral-of-its-quanti) and [here](http://math.stackexchange.com/questions/64186/intuition-behind-using-complementary-cdf-to-compute-expectation-for-nonnegative) and [here](http://math.stackexchange.com/questions/64186/intuition-behind-using-complementary-cdf-to-compute-expectation-for-nonnegative). I've only heard it expressed as 'the expectation is the integral of the survival function' ... rather than with any particular name. – Glen_b Oct 18 '13 at 20:04
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    Incidentally, for continuous random variables, you can show it in two lines using integration by parts. – Glen_b Oct 18 '13 at 20:14

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