> [!tldr] > The **log-normal distribution** is essentially the Gaussian exponentiated: if $Z \sim N(\mu, \sigma^{2})$, then $S:= \exp(Z) \sim \mathrm{LogNormal}(\mu, \sigma^{2}).$ ### Basic Properties The log-normal RV defined above has: $\begin{align*} \mathbb{E}[S]&= \exp\left( \mu +{ \frac{\sigma^{2}}{2}} \right),\\ \mathrm{Var}[S] &= [\exp(\sigma^{2}) -1 ] \cdot\underbrace{\exp(2\mu + \sigma^{2})}_{\text{expectation}^2}.\end{align*} $ ### Deriving the Distribution and Density Since the exponentiation is monotone, the following events are equivalent: $S \le s \iff Z \le \log(s).$Therefore, we can derive the distribution (cdf) of $S$ from that of the Gaussian: $\begin{align*} F_{S}(s)&:=\mathbb{P}[S \le s]\\ &~= \mathbb{P}[Z \le \log(s)]\\ &~= \mathbb{P}\left[ N(0,1) \le \frac{\log(s)-\mu}{\sigma} \right]\\ &= \Phi\left( \frac{\log(s)-\mu}{\sigma} \right), \end{align*}$where $\Phi$ is the standard Gaussian cdf. The density comes from taking the derivative of the above with respect to s: $\begin{align*} f_{S}(s)&= \frac{ \partial F_{S} }{ \partial s }\\ &= \frac{1}{s\cdot\sigma} \phi\left( \frac{\log(s)-\mu}{\sigma} \right)\\ &= \frac{1}{s} \phi(\log s; \mu, \sigma^{2}) \\ &= \frac{1}{\sqrt{ 2\pi \sigma^{2} }\cdot s} \exp\left[ -\frac{(\log(s)-\mu)^{2}}{2\sigma^{2}} \right] \end{align*}$where $\phi$ is the standard Gaussian density (pdf), and $\phi(\cdot ~; ~\mu, \sigma^{2})$ is that of $N(\mu, \sigma^{2})$.