How to calculate $ \mathbb{E}\left[X|W=0\right] $

Question

Let $ X\sim\mathcal{N}\left(0,1\right) $ and define $$ W=\begin{cases} 0 & X<0\\ X & X\geq0 \end{cases} = X\boldsymbol{1}_{X\geq0} $$

How can I calculate $ \mathbb{E}\left[X|W\right]$ ?

Here's what I have tried so far:

I want to identify this random variable for any value $ w $ such that $W=w$. So I first noticed that

And now all I have to do is to calculate $ \mathbb{E}\left[X|W=0\right] $. Since $W$ is neither a continuous or a discrete time variable, Im not sure how to express the conditioned density. Any help would be appreciated.

Thanks in advance.

Answer 1

$W=0$ iff $X \leq 0$. So $E(X|W=0)=E(X|X\leq 0)$. By symmetry of $N(0,1)$ we have $E(X|X>0)=E(-X|X <0)$. Also $E|X|=E(XI_{X>0})+E(XI_{X<0})$. Combine these to get $E(X|W=0)=\frac 1 2 E|X|$ and $E|X|=\sqrt {\frac 2 {\pi}}$