# If $X \sim U(-2,2)$ ,find the correlation between $X,|X|$

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?

• The correlation is indeed $0$. – Kavi Rama Murthy Feb 5 at 10:19
• Is my process correct? – Legend Killer Feb 5 at 10:20
• It is correct. In fact $X|X|$ itself has a symmetric distribution. – Kavi Rama Murthy Feb 5 at 10:21
• $E(X\,|X|)=E(X^2\mathbf1_{X>0})-E(X^2\mathbf1_{X<0})=0$, because $X$ is symmetric about zero. – StubbornAtom Feb 5 at 10:48
• 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 – Henry Feb 5 at 10:58