Log-Normal Distribution: Definition, Calculation, and Applications

A comprehensive guide to understanding the log-normal distribution, its definition, calculation methods, and real-world applications in statistics and beyond.

Definition

A log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. If \( X \) is a random variable with a normal distribution, then \( Y = e^X \) has a log-normal distribution, characterized by its skewness and limited range of positive values. Mathematically, if \( X \sim N(\mu, \sigma^2) \), then \( Y = e^X \sim \text{LogNormal}(\mu, \sigma^2) \).

$$ Y = e^X $$

Historical Context

The log-normal distribution was first introduced by the Scottish mathematician Robert G. Campbell in the early 20th century. It gained prominence in various fields for modeling skewed data, particularly when values must be positive.

Calculation Methods

Probability Density Function (PDF)

The probability density function (PDF) of a log-normal distribution is given by:

$$ f_Y(y; \mu, \sigma) = \frac{1}{y\sigma\sqrt{2\pi}} \exp \left( -\frac{(\ln y - \mu)^2}{2\sigma^2} \right) $$

where:

• \( y \) is the value of the random variable
• \( \mu \) is the mean of the natural logarithm of the variable
• \( \sigma \) is the standard deviation of the natural logarithm of the variable

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) is:

$$ F_Y(y ; \mu, \sigma) = \frac{1}{2} + \frac{1}{2} \text{erf} \left( \frac{\ln y - \mu}{\sigma \sqrt{2}} \right) $$

where \( \text{erf} \) is the error function.

Moments

The mean, variance, and other moments of a log-normal distribution can be derived as follows:

• Mean: \( E[Y] = e^{\mu + \frac{\sigma^2}{2}} \)
• Variance: \( \text{Var}(Y) = (e^{\sigma^2} - 1)e^{2\mu + \sigma^2} \)

Applications

Finance and Economics

Log-normal distributions are extensively used in finance to model asset prices, stock prices, and risk assessment. The Black-Scholes option pricing model, for instance, assumes that the underlying asset prices follow a log-normal distribution.

Environmental Studies

They are also used to model concentrations of pollutants, sizes of organisms within a species, and other naturally occurring phenomena that are positively skewed.

Reliability Engineering

In reliability engineering, the log-normal distribution is applied to model life durations of products and materials.

Real-World Examples

Example 1: Stock Prices

Assume that the logarithm of a stock price follows a normal distribution with a mean \(\mu = 0\) and standard deviation \(\sigma = 0.1\).

Example 2: Environmental Science

Consider the distribution of pollutant concentrations in a water body, where the pollutant levels are positively skewed.

Special Considerations

While the log-normal distribution is versatile, it assumes that data must be positive and often requires transformation for interpretation and analysis.

Related Terms

• Normal Distribution: A probability distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
• Black-Scholes Model: A mathematical model used for pricing options that assumes the log-normal distribution for stock prices.

FAQs

What are the key differences between a normal and a log-normal distribution?

A normal distribution is symmetric and defined for all real numbers, while a log-normal distribution is positively skewed and defined only for positive real numbers.

How do you transform data to fit a log-normal distribution?

Take the natural logarithm of the data. If the transformed data is normally distributed, then the original data follows a log-normal distribution.

Summary

The log-normal distribution is a crucial concept in statistics, finance, and many other fields. It models positively skewed data where values are constrained to be positive, providing a robust framework for various applications, from stock prices to environmental pollutant levels.

References

1. Aitchison, J., Brown, J. A. C. (1957). The Lognormal Distribution. Cambridge University Press.
2. Hull, J. C. (2018). Options, Futures, and Other Derivatives. Pearson Education.

This article aimed to provide a comprehensive guide on the log-normal distribution, covering its definition, calculation methods, and various applications. By understanding this distribution, readers can better analyze and interpret positively skewed data.