If $X \sim U[-1,1]$, show that $X$ and $X^2$ are uncorrelated but not independent

Question

$X = $Unif[-1,1] RV. Show X and $X^2$ are uncorrelated but not independent

I think I did something wrong cause $E[X^3]=E[X]E[X^2]$.
$f_X(x)=\frac{1}{1-(-1)}=\frac1{2}$

$E[X]=\int_{-\infty}^{\infty} xf_X(x)dx$

$E[X\times X^2]=E[X^3]=\int_{-1}^1\frac{x}2\frac{x^2}4dx=\frac18-\frac18$
$E[X]=\int_{-1}^1\frac{x}2dx=0$
$E[X^2]=\int_{-1}^1\frac{x^2}4=\frac{1}{12}+\frac{1}{12}=\frac1{6}$

What did I do wrong?

Answer 1

Use the comment above to make some corrections. However it is true that $EXEX^{2}=EX^{3}$. What this says is that the covariance of $X$ and $X^{2}$ is $0$ (as you were required to prove). This does not imply that $X$ and $X^{2} $ are independent. To prove that they are not independent assume that they are and get a contradiction. If they are independent the so are $X^{2}$ and $X^{2}$. (In fact $f(X)$ and $X^{2}$ would be independent for any measurable function $f$). But then $EX^{2}EX^{2}=EX^{4}$ You can see that this is not satisfied.