Definition:Gaussian Distribution

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

Then $X$ has a Gaussian distribution the probability density function of $X$ is:


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

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

This is written:


 * $X \sim N \paren {\mu, \sigma^2}$

Also known as
The Gaussian distribution is also commonly known as the normal distribution.

Both terms may be found on.

Also see

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