# Definition: ex-Gaussian distribution

**Index:**The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ ex-Gaussian distribution ▷ Definition

**Definition:** Let $A$ be a random variable that is normally distributed with mean $\mu$ and variance $\sigma^2$, and let $B$ be a random variable that is exponentially distributed with rate $\lambda$. Suppose further that $A$ and $B$ are independent. Then the sum $X=A+B$ is said to have an exponentially-modified Gaussian (i.e., ex-Gaussian) distribution, with parameters $\mu$, $\sigma$, and $\lambda$; that is,

where $\mu \in \mathbb{R}$, $\sigma>0$, and $\lambda > 0$.

**Sources:**

- Luce, R. D. (1986): "Response Times: Their Role in Inferring Elementary Mental Organization", 35-36; URL: https://global.oup.com/academic/product/response-times-9780195036428.

**Metadata:**ID: D187 | shortcut: exg | author: tomfaulkenberry | date: 2023-04-18, 12:00.