\begin{definition}[Definition:Gaussian Distribution]
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ has a '''Gaussian distribution''' {{iff}} 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: 
:$X \sim \Gaussian \mu {\sigma^2}$
\end{definition}