# Definition:Gaussian Distribution

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## Definition

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Then $X$ has a **Gaussian distribution** if and only if the probability density function of $X$ is:

- $\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \, \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$

for $\mu \in \R, \sigma \in \R_{> 0}$.

This is written:

- $X \sim \Gaussian \mu {\sigma^2}$

## Also known as

The **Gaussian distribution** is also commonly known as the **normal distribution** (hence the notation).

The former term is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Expectation of Gaussian Distribution: $\expect X = \mu$
- Variance of Gaussian Distribution: $\var X = \sigma^2$

- Results about
**the Gaussian distribution**can be found here.

## Source of Name

This entry was named for Carl Friedrich Gauss.

## Technical Note

The $\LaTeX$ code for \(\Gaussian {\mu} {\sigma^2}\) is `\Gaussian {\mu} {\sigma^2}`

.

When either argument is a single character, it is usual to omit the braces:

`\Gaussian \mu {\sigma^2}`

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Gaussian curvature** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($1777$ – $1855$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Gaussian distribution** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Gaussian distribution** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Gaussian distribution**

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 39$: Probability Distributions: Normal Distribution: $39.4$