\[f(x)=\frac{1}{x\sigma\sqrt{2\pi}}e^{-\frac{(\ln(x)-\mu)^2}{2\sigma^2}}\]
Variables
fprobability density
xrandom variable value
mumean of the logarithmic values
sigmastandard deviation of the logarithmic values
Description
What is this formula?
This formula calculates the probability density of a lognormal distribution. A variable follows a lognormal distribution when its logarithm is normally distributed.
When to use it
Use this formula for positive-valued variables with asymmetric distributions, especially when values grow multiplicatively.
Example
The formula can model particle sizes, stock prices, biological growth, or income distributions where small values are common and large values are rare.
Applications
Finance, reliability engineering, environmental analysis, economics, telecommunications, hydrology, and natural growth processes.