📂Probability Distribution

Log-Normal Distribution

Definition ¹

The continuous probability distribution $\log N \left( \mu,\sigma^{2} \right)$, which has the probability density function given for $\mu \in \mathbb{R}$ and $\sigma^{2} > 0$, is known as the log-normal distribution. $$ f(x) = {{ 1 } \over { x \sigma \sqrt{2 \pi}}} \exp \left[ - {{ \left( \log x - \mu \right)^{2} } \over { 2 \sigma^{2} }} \right] \qquad, x > 0 $$

Description

In fact, the above definition is ridiculously complicated, and intuitively, a random variable $X$ that follows a normal distribution when the log function is applied, is said to have a log-normal distribution. $$ \log X \sim N \left( \mu , \sigma^{2} \right) $$ Occasionally, it is also referred to as the Galton distribution, named after the eugenicist Francis Galton.

A notable application of the log-normal distribution is in geometric Brownian motion.