Expected value of $|X|$ given $X$ when $X$ is standard normal

Question

Considering $X\sim N(0,1)$, I would like to know why

$$E\left[|X|\mid X \right] = |X|$$

I tried to prove that with the definition of expected value, but I've failed. I would like to see the formal calculation/proof for that. enter image description here

Answer 1

Recall the general rule, that if $\varphi(x) = \mathbb{E}[Y \: | \: X=x]$, then $\varphi(X) = \mathbb{E}[Y \: | \: X]$.

It should be obvious, that $P(|X|=|x| \: | \: X = x) = 1$, and therefore $$ \mathbb{E}[|X| \: | \: X=x] =|x|,$$ from which we conclude that $\mathbb{E}[|X| \: | \: X] = |X|$.