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Does anyone know a example of a random variable $X:\Omega \to \mathbb{R}$ which $$E[X] \in \mathbb{R}$$ but $$E[X^{2}] \not\in \mathbb{R}\:?$$

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    $\begingroup$ Take $P(X=n)=\frac{1}{\zeta(3)n^3}$ for each $n \geq 1$. $\endgroup$ – Mindlack Dec 30 '20 at 18:41
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    $\begingroup$ Does this answer your question: math.stackexchange.com/questions/236181/…? $\endgroup$ – Physical Mathematics Dec 30 '20 at 18:44
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    $\begingroup$ $c/(1+|x|^3)$ where $c$ is a normalizing factor. $\endgroup$ – Chrystomath Dec 30 '20 at 18:44
  • $\begingroup$ $X(\omega) = {1 \over \sqrt{\omega}}$ on $(0,\infty)$. $\endgroup$ – copper.hat Dec 30 '20 at 19:10
  • $\begingroup$ Sorry, that should be $(0,1)$ with uniform distribution. $\endgroup$ – copper.hat Dec 30 '20 at 19:17

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