What is the conditional expectation of $E(X|X<0)$ where $X$ is normally distributed with mean $0$?

Question

I am trying to solve another problem on conditional expectation and have boiled it down to this problem: What is the conditional expectation of $E(X|X<0)$ where $X$ is normally distributed with mean $0$? Any hint or suggestion how to solve this problem? I have seen on the wikipedia page that the expected value of the half normal distribution is $\frac{\sigma \sqrt{2}}{\sqrt{\pi}}$, hence my guess is that $E(X|X<0) = - \frac{\sigma \sqrt{2}}{\sqrt{\pi}}$, Am I correct with this guess?

Answer 1

this might help you. $E[X|X<0]=\frac{\int_{-\infty}^0xp_{X}(x)dx}{\int_{-\infty}^0 p_{X}(x)dx}$ where $p_{X}(x)$ is the pdf of random variable $X$.