Applying Conditional Expectation Formula

Question

Suppose $X$ is a random variable, we want to find $\mathbb{E}(X\ |\ X>0)$. Is that equivalent to $\frac{\mathbb{E}(X\cdot \mathbb{1}_{X>0})}{\mathbb{P}(X>0)}$? From what I've seen online, it says that $$\mathbb{E}(X\ |\ H) = \frac{\mathbb{E}(X \mathbb{1}_H)}{\mathbb{P}(H)}$$ where $H\in \mathcal{F}$ is some event in the $\sigma$-algebra. Thus, does this apply to my specific case? Additionally, if somebody could explain why the above formula works, that'd be great as well. Thanks.

Answer 1

Yes, $X>0$ describes an event in the sigma-algebra generated by random variable $X$.

Thus indeed:$$\mathbb E(X\mid X>0)~=~\dfrac{\mathbb E(X\,\mathbf 1_{X>0})}{\mathbb P(X>0)}\raise{0.5ex}.$$

It is the expected value for $X$ measured over the event of $X>0$.