Uncorrelated but not independent random variable

Question

I have a uniform discrete RV defined as $$X \sim Uni \{-1, 1\}$$. Now, I want to create another uniform random variable from $X$ defined as $Y =X^2$.

Am I right to say the following?

1. $$E[XY] -E[X]E[Y] =E[X^3] - E[X] =0$$
2. $X$ and $Y$ are uncorrelated.
3. $X $ and $ Y$ are dependent

I want someone to help me confirm this. Much appreciated.

Answer 1

Consider any random variable $X$ taking values in the set $\{-1,0,1\}$ and has a PMF satisfying: $p_X(-1) = p_X(1) = \rho \in (0,0.5)$.

$\text{Cov}(X,X^2)=\mathbb{E}(X^3) - \mathbb{E}(X)\mathbb{E}(X^2) = 0 - 0\times 2\rho = 0$.

So, $X$ and $X^2$ are uncorrelated. However, these are not independent because $\Pr(X=0, X^2=0) = \Pr(X=0) = 1-2\rho \neq (1-2\rho)^2 =\Pr(X=0)\Pr(X^2=0) $