A probability distribution on a Cartesian space $\mathbb{R}^n$ is called Gaussian or a normal distribution if it is of the form
where $A$ is some $n \times n$ matrix such that $\langle -, A-\rangle$ is a positive definite bilinear form. Here $\det A$ denotes the determinant and $\langle -,-\rangle$ is the canonical bilinear form on $\mathbb{R}^n$.
Since $\sqrt{\det A}$ is the coordinate of the volume element $vol_A$ associated with $A$, we may equivalently write this as
The mean of this distribution is $\vec{0}$; for a distribution with mean $\vec{c}$, replace $\langle{\vec{x},A \vec{x}}\rangle$ with $\langle{\vec{x} - \vec{c},A \vec{x} - A \vec{c}}\rangle$.
The matrix $A$ is the inverse of the covariance matrix?. In particular, for $n = 1$, we may write $x^2/\sigma^2$ (or $(x-c)^2/\sigma^2$ for mean $c$) in place of $\langle{\vec{x},A \vec{x}}\rangle$, where $\sigma$ is the standard deviation; similarly, $\sqrt{\det A}$ becomes $1/\sigma$. This gives the form
which may be more familiar to some readers.
See also
Wikipedia, Normal distribution
Wikipedia, Standard deviation
Last revised on January 17, 2019 at 12:28:34. See the history of this page for a list of all contributions to it.