# 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

• 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}