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
Good Calculators, Standard deviation calculator
Last revised on December 23, 2020 at 08:56:35. See the history of this page for a list of all contributions to it.