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If I know the $\mathbb{E}(X)$ and $\mathbb{E}(X^2)$ of some random variable $X$, can I get $\mathbb{E}(|X|)$? Or are there any bounds related to $\mathbb{E}(|X|)$?

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$ |\mathrm E(X)|\leqslant\mathrm E(|X|)\leqslant\sqrt{\mathrm E(X^2)} $ Edit: As suggested by @Qiaochu, for every $(a,b,c)$ such that $a^2\leqslant b^2\leqslant c^2$, there exists a Bernoulli random variable $X$ such that $\mathrm E(X)=a$, $\mathrm E(|X|)=b$ and $\mathrm E(X^2)=c^2$. That is, there exists $p$ in $[0,1]$ and $(x,y)$ such that the distribution $\mathrm P(X=x)=p$, $\mathrm P(X=y)=1-p$, is a solution.

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    And I think you can't say more than this in general (that is for any triple of real numbers satisfying this chain of inequalities you can find a corresponding $X$, and in fact I think you can do this with a finite sample space).2012-01-06