Basics of Gaussian Distribution

For a single variable $x$ that follows a Gaussian distribution, we have the probability density function (pdf) as follows:

\[\begin{equation} p(x|\mu, \sigma) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \end{equation}\]

For a $D$-dimensional vector $\bm{x}$, the pdf is:

\[\begin{equation} p(\x|\bm{\mu}, \bm{\Sigma}) = \frac{1}{(2\pi)^{D/2}|\bm{\Sigma}|^{1/2}} \exp\nbr{-\frac{1}{2}(\bm{x}-\bm{\mu})^T\bm{\Sigma}^{-1}(\bm{x}-\bm{\mu})} \end{equation}\]

where $\bm{\mu}$ is a $D$-dimensional mean vector and $\bm{\Sigma}$ is a $D\times D$ covariance matrix.