多変量正規分布
\(\boldsymbol{X}=(X_1, X_2, …, X_n)^{\mathsf{T}}\)が,以下の同時確率密度関数に従うとき,\(\boldsymbol{X}\)を多変量正規分布と言う. \[ \boldsymbol{\mu} = (\mu_1, \mu_2, …, \mu_n)^{\mathsf{T}} \\ \sigma_{ij} = \mathrm{Cov}(X_i, X_j) \\ \boldsymbol{\Sigma} = \begin{pmatrix} \sigma_{11} & \sigma_{12} & \cdots & \sigma_{1n} \\ \sigma_{21} & \sigma_{22} & \cdots & \sigma_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{n1} & \sigma_{n2} & \cdots & \sigma_{nn} \\ \end{pmatrix} \\ \] として,
\[ f_{\boldsymbol{X}}(\boldsymbol{x}|\boldsymbol{\mu},\boldsymbol{\Sigma})=\Bigr(\frac{1}{2\pi}\Bigl)^{n/2}\frac{1}{|\boldsymbol{\Sigma}|^{1/2}}\exp\Bigl\{ -\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu})^{\mathsf{T}}\boldsymbol{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu}) \Bigr\} \]