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:
Definition of 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.
Summary Table
| 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) |
In Finance
- Stock returns (log returns) are often modeled as normal.
- Stock prices, which are the exponentials of those returns, follow a lognormal distribution.
Why the Link Matters
This relationship allows:
- Analytical pricing of options (e.g., Black-Scholes assumes lognormal prices).
- Modeling multiplicative effects (lognormal) by transforming them into additive (normal) problems.