【系列二】数学基础—概率—高斯分布2

1. 高维(二维)高斯分布

$$x_i\backsim N(\mu ,\Sigma )=\frac{1}{(2\pi {)}^{\frac{p}{2}}{\Sigma }^{\frac{1}{2}}}\exp (-\frac{1}{2}(x-\mu {)}^T{\Sigma }^{-1}(x-\mu ))$$
x = ( x 1 x 2 ⋮ x p ) , μ = ( μ 1 μ 2 ⋮ μ p ) , Σ = ( σ 11 , σ 12 … , σ 1 p σ 21 , σ 22 … , σ 2 p ⋮ σ p 1 , σ p 2 … , σ p p ) x=\begin{pmatrix}x_1\\x_2\\\vdots\\x_p\end{pmatrix},\mu=\begin{pmatrix}\mu_1\\\mu_2\\\vdots\\\mu_p\end{pmatrix},\varSigma=\begin{pmatrix}\sigma_{11},\sigma_{12}\dotsc,\sigma_{1p}\\\sigma_{21},\sigma_{22}\dotsc,\sigma_{2p}\\ \vdots\\\sigma_{p1},\sigma_{p2}\dotsc,\sigma_{pp}\end{pmatrix} x=⎝⎜⎜⎜⎛​x1​x2​⋮xp​​⎠⎟⎟⎟⎞​,μ=⎝⎜⎜⎜⎛​μ1​μ2​⋮μp​​⎠⎟</