Log-normal distribution

A positive random variable is log-normal when its logarithm has a normal distribution. If $Z\sim N(0,1)$ and $X=e^Z$, then $\mathbb{E}[X^n]=e^{n^2/2}$.

The support is $x>\theta$ and the probability density function isThe change of variables $z=\log (x-\theta )$ proves normalization. The threshold $\theta$ is a lower endpoint, distinct from the log-mean $m$ and log-standard-deviation $\sigma$. In measured sea-ice pressure ridge spacings, finite ridge widths and detection rules can create such a lower endpoint.

If $\log S=m+sZ$ with a standard normal $Z$, then $\mathbb{E}[S^p{1}_{\{S>K\}}]=e^{pm+p^2{s}^2/2}\Phi ((m+p{s}^2-\log K)/s)$. Completing the square proves the formula for every real $p$, including the inverse-price moment $p=-1$.