Brian Lee
See Normal Distribution Rule for typical ranges in a normal distribution.
The normal and lognormal distributions are closely related, but they describe very different kinds of random variables. Here’s a breakdown of their relationship:
If a random variable \(X\) is normally distributed, then the variable
\[ Y = e^X \]
is lognormally distributed.
Conversely, if \(Y\) is lognormally distributed, then
\[ \log Y \]
is normally distributed.
| Property | Normal Distribution | Lognormal Distribution |
|---|---|---|
| Variable | \(X \sim \mathcal{N}(\mu, \sigma^2)\) | \(Y = e^X \sim \text{Lognormal}(\mu, \sigma^2)\) |
| Support | \((-\infty, \infty)\) | \((0, \infty)\) |
| Shape | Symmetric bell curve | Skewed right |
| Mean | \(\mu\) | \(e^{\mu + \sigma^2/2}\) |
| Median | \(\mu\) | \(e^\mu\) |
| Mode | \(\mu\) | \(e^{\mu - \sigma^2}\) |
| Use case example | Returns (log change) | Prices or quantities (positive only) |
This relationship allows: