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If $X \sim U(-2,2)$ ,find the correlation between $X,|X|$ It is clear that $|X| \sim U(0,2)$. Obviously , $E(X)=0$ Then $E(X|X|)=E(X^2 \cap X>0)-E(X^2 \cap X \le 0)=0$ due to symmetry. Thus, the correlation coefficient becomes 0.Am I going wrong somewhere?

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    $\begingroup$ The correlation is indeed $0$. $\endgroup$ – Kavi Rama Murthy Feb 5 at 10:19
  • $\begingroup$ Is my process correct? $\endgroup$ – Legend Killer Feb 5 at 10:20
  • $\begingroup$ It is correct. In fact $X|X|$ itself has a symmetric distribution. $\endgroup$ – Kavi Rama Murthy Feb 5 at 10:21
  • $\begingroup$ $E(X\,|X|)=E(X^2\mathbf1_{X>0})-E(X^2\mathbf1_{X<0})=0$, because $X$ is symmetric about zero. $\endgroup$ – StubbornAtom Feb 5 at 10:48
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    $\begingroup$ Implicitly you are saying the covariance is $E[X\,|X|] - E[X]E[|X|]$ and that both $E[X\,|X|]=0$ and $E[X]=0$, so the covariance is $0$ and thus the correlation is $0$. It might be worth saying this explicitly $\endgroup$ – Henry Feb 5 at 10:58

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