I have a variable $\omega$ that is lognormal, that is, when we take the log of $\omega$ the distribution is normal. I saw in a paper the following result:
$$ E\left[\omega | \omega\geq \bar{\omega} \right] = \dfrac{1-\Phi(z-\sigma)}{1-\Phi(z)}$$
where $z = \dfrac{log(\bar{\omega}) + 0.5\sigma^2}{\sigma}$ and where $\Phi(z)$ denotes the CDF of the standard normal variable $z$. Also, $\log(\bar{\omega}) \sim N(-\dfrac{1}{2}\sigma^2, \sigma^2)$
How can you reach to the conditional expectation shown above? I am more familiar with this:
$$E\left[\omega | \omega\geq \bar{\omega} \right] = \int_{\bar{\omega}}^{\infty}\dfrac{\omega f(\omega) d\omega}{1-F(\omega)}$$
How can this be re-expressed in terms of CDFs as above?
The result shown above is here in page 1385