Definition:Gaussian Distribution

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

Then $X$ has a Gaussian distribution 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.

Also see

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