Example for uncorrelated but not independent random variables

Question

Let $X$ be a standard normal random variable and define $$ Y = \begin{cases} X, &\mbox{if } |X| \leq c, \\ -X, & \mbox{if } |X| > c . \end{cases} $$ The article Normally distributed and uncorrelated does not imply independent says that it is possible to choose $c > 0$ such that $X$ and $Y$ are uncorrelated. How can I find this constant $c$?

Answer 1

HINT $$ \mathbb{E}\left[X^2 \mathbb{I}_\left\{|X| \le c\right\}\right] = \int_{x=-c}^{x=c} x^2 \phi(x)dx, $$ where $\phi(x)$ is the pdf of the standard normal.

You can do the same to the other integral and find $c$ numerically.