Table of Contents
What Is the Lognormal Distribution?
A random variable X follows a lognormal distribution if its natural logarithm ln(X) is normally distributed. The lognormal distribution is always positive and right-skewed, making it suitable for modeling quantities that cannot be negative.
It arises naturally in many contexts: stock prices (multiplicative random walks), income distributions, particle sizes, biological measurements, and failure times. Any process where the variable grows by random percentage changes tends to produce lognormal data.
Formulas
Applications
| Field | Application |
|---|---|
| Finance | Stock prices, Black-Scholes model |
| Environmental | Pollutant concentrations |
| Engineering | Fatigue life, component reliability |
| Biology | Body weight, cell sizes |
Frequently Asked Questions
What are mu and sigma in the lognormal distribution?
μ and σ are NOT the mean and standard deviation of the lognormal distribution itself. They are the mean and standard deviation of the underlying normal distribution (i.e., of ln(X)). The actual mean and variance of the lognormal distribution are derived from these.
How does the lognormal relate to the normal distribution?
If X is lognormally distributed, then ln(X) is normally distributed. Conversely, if Y is normally distributed, then exp(Y) is lognormally distributed. This relationship allows using normal distribution tables for lognormal calculations.
When is the lognormal distribution appropriate?
Use it when your data is strictly positive, right-skewed, and arises from multiplicative processes. If taking the log of your data produces a bell-shaped (normal) distribution, the original data is likely lognormal.