How to show $\mathbb{E}(X) = \int_0^{\infty} (1-F_X(x)) \, dx$ for a continuous random variable $X \geq 0$ without assuming $f_X$ exists?

Asked Sep 27 '15 at 17:42

Active Sep 27 '15 at 17:42

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If $X$ is a continuous random variable taking non-negative values only, how do I show that $$\mathbb{E}(X)=\int_0^{\infty}[1-F_X(x)]dx$$

It's easy to give a proof if we assume the density function $f_X$ exists (use Fubini's Theorem to interchange the integrations). However, I got stuck in the general case. What if $X$ does not have a density function?

asked Sep 27 '15 at 17:42

zxzx179

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"X has a density function" is usually the definition of "X is a continuous random variable". – Nate Eldredge Sep 27 '15 at 17:47
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@NateEldredge: The definition I know only requires that the distribution of $X$ assigns zero measure to all singletons (i.e. the distribution function is continuous (not necessarily absolutely continuous)). – PhoemueX Sep 27 '15 at 19:09
Yeah, an example of a continuous CDF with no PDF is the Cantor staircase. – Did Sep 27 '15 at 19:54

How to show $\mathbb{E}(X) = \int_0^{\infty} (1-F_X(x)) \, dx$ for a continuous random variable $X \geq 0$ without assuming $f_X$ exists?

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