Finding $E[X\;|\; Y]$ that $Y=|X|$ in $N(0,1)$

Question

Suppose $X\sim N(0,1)$. How can calculate $E[X\;|\; Y]$ that $Y=|X|$ please guide me.

Answer 1

It is intuitevely expected that, given the value of $|X|$, the value of $X$ is $|X|$ or $-|X|$ with equiprobability. This implies $E[X \mid |X|]=0$.

To show the first point, consider a random variable $Z \sim {\cal N}(0,1)$ and a $(\frac12,\frac12)$-Bernoulli random variable $\epsilon$ independent of $Z$, and define a new random variable $X$ by setting $$X= \begin{cases} |Z| & \text{if } \epsilon=0 \\ -|Z| & \text{if } \epsilon=1 \end{cases}.$$

Thus, clearly, the value of $X$ given the value of $|Z|$ is $|Z|$ or $-|Z|$ with equiprobability. Then check that $X \sim {\cal N}(0,1)$ and $|X|=|Z|$.