Definition: Log-normal distribution

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ Log-normal distribution ▷ Definition

Definition: Let $\ln X$ be a random variable following a normal distribution with mean $\mu$ and variance $\sigma^2$ (or, standard deviation $\sigma$):

\[\label{eq:norm} Y = \ln (X) \sim \mathcal{N}(\mu, \sigma^2) \; .\]

Then, the exponential function of $Y$ is said to have a log-normal distribution with location parameter $\mu$ and scale parameter $\sigma$

\[\label{eq:lognorm} \begin{split} X = \mathrm{exp}(Y) \sim \ln \mathcal{N}(\mu, \sigma^2) \end{split}\]

where $\mu \in \mathbb{R}$ and $\sigma^2 > 0$.

 

Sources:

• Wikipedia (2022): "Log-normal distribution"; in: Wikipedia, the free encyclopedia, retrieved on 2022-02-07; URL: https://en.wikipedia.org/wiki/Log-normal_distribution.

Metadata: ID: D170 | shortcut: lognorm | author: majapavlo | date: 2022-02-07, 22:33.