Independence between |X| and sign(X)

Medium

Exercise 5 / 9

Let $X \sim \mathcal{N}(0, 1)$ . We define:

\mathrm{sign}(X) = \begin{cases} +1 & \text{if } X \geq 0 \\ -1 & \text{if } X < 0 \end{cases}

Question: Show that the random variables $|X|$ and $\mathrm{sign}(X)$ are independent.