Dependency and correlation between a random variable and it's square I have the following question. At first this seemed very silly but after thinking about it, I found my self struggling.
Given $X$, a random variable, I should decide if the following sentences are right or wrong and explain why:
1 - $X$ and $X^2$ are always independent.
2 - $X$ and $X^2$ are never independent.
3 - $X$ and $X^2$ are always correlated.
4 - $X$ and $X^2$ are never correlated.
Here are my answers. I was able (I think) to answer dependency part, but the correlation part I couldn't.
1 - False: since $X^2$ could only be achieved with $X$ and $-X$, it is dependat with $X$.
2 - False: As said before, we can get $X^2$ using $-X$. This way $X^2$ and $X$ are independant.
3 & 4 I don't have a clue.
Could you guys help me with it? Did I answer questions 1 and 2 right? any explenation by words or via mathemtical approach would help.
Thank you!

Just to spell it out in more detail than the comments, here's one counterexample for 3. Let $ X \sim U[-1,1] $ and let $ Y = X^2 $. Then $$ \begin{align*}
\mathbb{C}\text{ov}(X,Y) &= \mathbb{E}[X^3] \\
&= \int_{-1}^1 \frac{x^3}{2} dx \\
&= 0.
\end{align*}
$$
For question 1, you can basically use any example where squaring would lead to a constant. Remember that a constant is independent of all random variables, including itself.
