Factsheet: Lognormal distribution
Where to use: The lognormal distribution is used to model continuous random variables with values that are both real and non-negative, wherein the logarithms of these variables follow a normal distribution. That is to say, if the random variable \(X\) is lognormally distributed, then the random variable \(Y = \ln(X)\) is normally distributed (where \(\ln\) is the natural logarithm).
Notation: \(X \sim \textrm{Lognormal}(\mu,\sigma^2)\)
Parameters: As with the normal distribution, two numbers \(\mu\) and \(\sigma^2\) where:
- \(\mu\) is the expected value of the normally distributed random variable \(Y = \ln(X)\),
- \(\sigma^2\) is the variance of the normally distributed random variable \(Y = \ln(X)\).
| Quantity | Value | Notes |
|---|---|---|
| Mean | \(\mathbb{E}(X) = \exp(\mu+\frac{\sigma^2}{2})\) | \(\exp(y) = e^y\) |
| Variance | \(\mathbb{V}(X) = [\exp(\sigma^2)-1]\exp(2\mu+\sigma^2)\) | \(\exp(y) = e^y\) |
| \(\mathbb{P}(X=x)=\dfrac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\dfrac{(\ln(x)-\mu)^2}{2\sigma^2}\right)\) | \(\exp(y) = e^y\) | |
| CDF | \(\displaystyle\mathbb{P}(X\leq x)=\dfrac{1}{2}\left[1+\textrm{erf}\left(\dfrac{\ln(x)-\mu}{\sigma\sqrt{2}}\right)\right]\) | \(\textrm{erf}(x)\) is the error function of \(x\) |
Example: The logarithms of Cantor’s Confectionery’s stock prices follow a normal distribution. The mean of the stock prices’ natural logarithms is \(8.01\), whereas the variance of the stock prices’ natural logarithms is \(3\). This can be expressed as \(X \sim \textrm{Lognormal}(8.01, 3)\), meaning the logarithm of the location parameter is \(8.01\) and the logarithm of scale parameter is \(3\).
Further reading
Version history
v1.0: initial version created 04/25 by tdhc and Michelle Arnetta as part of a University of St Andrews VIP project.
v1.1: moved to factsheet form and populated with material from Overview: Probability distributions by tdhc.
v1.2: deprecated shinylive apps replaced 09/26 by tdhc.