# An inequality for the non-negative random variable

I once read the following inequality

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

• 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
• 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
• See discussion of the result here and here and here. 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
• Incidentally, for continuous random variables, you can show it in two lines using integration by parts. – Glen_b Oct 18 '13 at 20:14