Are uniformly distributed random variables $X$, $X^2$ (in)dependent?

Question

We have a random variable $X \sim U[-1,1]$. I should find out whether $X, X^2$ are independent or not. I have shown that $\textrm{cov}(X,Y) = 0$ and that PDFs are $f_X(x) = \frac{1}{2} $ for $x \in [-1,1]$ and $0 $ else and $f_{X^2}(x) = \frac{1}{2\sqrt{x}}$ for $x \in [0,1]$ and $0$ else.

However I am not sure how to show that they are (in)dependent.

Answer 1

Hint:

\begin{align} P(X^2 \le \frac14, -\frac12 \le X \le \frac12) &= P( -\frac12 \le X \le \frac12)= \frac12 \end{align}

Answer 2

They are dependent, since knowing the value of one variable updates your probability distribution for the other. To know $X$ is to know $X^2$.

Answer 3

For proving that $X$ and $X^2$ are not independent it is enough to find Borel set $A,B$ such that:$$P(X\in A,X^2\in B)\neq P(X\in A)P(X^2\in B)$$ Give it a try for e.g. $A=[\frac12,1]$ and $B=[\frac14,1]$.