question about the notation of conditional expectation

Question

Let $X$ be a standard normal random variable. We need to compute the integral $$E(X~|X>0).$$ In the book, they give the answer ${1\over \sqrt{2\pi}}$. Then I am confused with this conditional expectation notation. Since from my understanding, it should be $$\int_0^\infty 2\cdot {x\over \sqrt{2\pi}} e^{-{x^2\over 2}}\, dx$$ which gives the answer $\sqrt{{2\over\pi}}$. I add a factor "2" here. Can anyone tell me this conditon expectation $E(X|X>0)$ notation meaning? Since it is not standard, we usually conditioning on a $\sigma$-algebra, rather than a set.

Answer 1

By the wikipedia page for conditional expectation, we should have $$E(X|A) = \frac{E(X\mathbf{1}_A)}{P(A)}$$

In particular, $E(X|X>0) = \sqrt{\frac{2}{\pi}}$ when $X$ is a standard normal.

---

Related, I believe I have seen the notation $E(X;A) := E(X\mathbf{1}_A),$ which would give an answer of $E(X;X>0) = \frac{1}{\sqrt{2\pi}}$