Lognormal Distribution

Study guide: Exam FAM

The Lognormal Distribution X \sim \text{Lognormal}(\mu, \sigma^2) applies when \ln X \sim N(\mu, \sigma^2). It is widely used to model insurance losses, asset prices, and any quantity that must be positive and right-skewed.

f(x) = \frac{1}{x\sigma\sqrt{2\pi}}\exp\!\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right)

x > 0

\text{where } \mu \in \mathbb{R} = \text{log-mean},\; \sigma > 0 = \text{log-standard deviation}

E[X] = e^{\mu + \sigma^2/2}

\text{Var}(X) = e^{2\mu+\sigma^2}(e^{\sigma^2}-1)

F(x) = \Phi\!\left(\frac{\ln x - \mu}{\sigma}\right)

Probability a Loss Exceeds a Threshold Losses follow X \sim \text{Lognormal}(\mu = 6,\, \sigma = 1.5). Find P(X > 1000).

Answer P(X > 1000) = 1 - \Phi\!\left(\frac{\ln 1000 - 6}{1.5}\right) = 1 - \Phi\!\left(\frac{6.908 - 6}{1.5}\right) = 1 - \Phi(0.605) \approx 1 - 0.7274 = 0.2726