Posts in Probability & Stats
Learning Machine Learning 2 - The multivariate Gaussian

If a $$N$$-dimensional random vector $$\mathbf{X}=(X_1,X_2,\dots,X_N)^\top$$ is distributed according to a multivariate Gaussian distribution with mean vector \begin{equation*}\boldsymbol{\mu}=(\Exp[X_1],\Exp[X_2],\dots,\Exp[X_N])^\top\end{equation*} and covariance matrix $$\mathbf{\Sigma}$$ such that $$\Sigma_{ij}=\cov(X_i,X_j)$$ we write $$\mathbf{X}\sim\mathcal{N}(\boldsymbol{\mu},\mathbf{\Sigma})$$ and the probability density function is given by \begin{equation*}p(\mathbf{x}|\boldsymbol{\mu},\mathbf{\Sigma})=\frac{1}{(2\pi)^{N/2}\sqrt{\det\mathbf{\Sigma}}}\exp\left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^\top\mathbf{\Sigma}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right).\end{equation*}