Show that for any continuous random variable $X$ that takes only positive real values $\int_{0}^{\infty}\text{Pr}(X\geq x)dx=\mu$ where $\mu$ is the mean.
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2This question has been discussed many times on this forum. See, for example, Intuition behind using complementary CDF to compute expectation for nonnegative random variables – Dilip Sarwate Jan 30 '15 at 14:25
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$\int_{0}^{+\infty}\text{Pr}(X\geq x) dx = \int_{0}^{+\infty} \mathbf{E}[\mathbf{1}_{\{X\geq x\}}] dx = \ldots$ What can you say after ?
Olórin
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