A log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. This means that if you take the natural logarithm of a log-normally distributed variable, the result will follow a normal distribution. Log-normal distributions are commonly used in various fields such as finance, environmental studies, and engineering, where values are positively skewed and cannot be negative.
In a log-normal distribution, the variable is defined as being greater than zero, which makes it suitable for modeling non-negative quantities. The distribution is characterized by two parameters: the mean (µ) and standard deviation (σ) of the variable’s natural logarithm. The probability density function (PDF) of a log-normal distribution is expressed as:
f(x; µ, σ) = (1 / (xσ√(2π))) * exp[-(ln(x) – µ)² / (2σ²)]
Where:
- x is the variable of interest.
- µ is the mean of the natural logarithm of the variable.
- σ is the standard deviation of the natural logarithm of the variable.
Log-normal distributions are particularly useful for modeling phenomena such as income distribution, stock prices, and the sizes of living organisms, where values tend to cluster around a central point but can take on a wide range of values. Understanding log-normal distributions helps in making predictions and assessments in various applications, particularly when dealing with multiplicative processes.