$f(x) = \frac{1}{x\,\sigma\sqrt{2\pi}}\,\exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right), \quad x > 0$
$\text{where } \mu \in \mathbb{R} = \text{log-mean},\quad \sigma > 0 = \text{log-standard deviation}$
The Lognormal Distribution $X \sim \text{Lognormal}(\mu, \sigma^2)$ is a continuous distribution on $(0, \infty)$ in which $\ln X \sim N(\mu, \sigma^2)$; it arises naturally when a quantity is the product of many independent positive factors.
Its mean is $E[X] = e^{\mu + \sigma^2/2}$ and variance is $\text{Var}(X) = e^{2\mu + \sigma^2}(e^{\sigma^2}-1)$. It is heavily right-skewed and widely used in actuarial science to model loss severity, as large losses are disproportionately likely compared to a normal distribution.
> [!example]- Expected Claim Size from a Lognormal Model {💡 Example}
> Claim sizes follow $X \sim \text{Lognormal}(\mu = 7, \sigma^2 = 1)$. Find $E[X]$ and $P(X > 5{,}000)$.
>
> > [!answer]- Answer
> > $E[X] = e^{7 + 1/2} = e^{7.5} \approx 1{,}808.04$
> > For the tail probability, note $\ln(5000) \approx 8.517$:
> > $P(X > 5000) = P\!\left(Z > \frac{8.517 - 7}{1}\right) = P(Z > 1.517) \approx 1 - 0.9353 = 0.0647$
> > About 6.5% of claims exceed \$5,000.