# 多元正态分布

（重定向自多变量正态分布

參數 Many samples from a multivariate (bivariate) Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.878, 0.478) direction (longer vector) and of 1 in the second direction (shorter vector, orthogonal to the longer vector). 概率多变量函數 μ ∈ RN — 位置 Σ ∈ RN×N — 协方差矩阵 (半正定) x ∈ μ+span(Σ) ⊆ RN ${\displaystyle (2\pi )^{-{\frac {N}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }})'{\boldsymbol {\Sigma }}^{-1}(\mathbf {x} -{\boldsymbol {\mu }})},}$ （仅当 Σ 为正定矩阵时） 解析表达式不存在 μ μ Σ ${\displaystyle {\frac {1}{2}}\ln((2\pi e)^{N}|{\boldsymbol {\Sigma }}|)}$ ${\displaystyle \exp \!{\Big (}{\boldsymbol {\mu }}'\mathbf {t} +{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} {\Big )}}$ ${\displaystyle \exp \!{\Big (}i{\boldsymbol {\mu }}'\mathbf {t} -{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} {\Big )}}$

## 一般形式

N维随机向量${\displaystyle \ X=[X_{1},\dots ,X_{N}]^{T}}$如果服从多变量正态分布，必须满足下面的三个等價条件：

• 任何线性组合${\displaystyle \ Y=a_{1}X_{1}+\cdots +a_{N}X_{N}}$服从正态分布
• 存在随机向量${\displaystyle \ Z=[Z_{1},\dots ,Z_{M}]^{T}}$( 它的每个元素服从独立标准正态分布），向量${\displaystyle \ \mu =[\mu _{1},\dots ,\mu _{N}]^{T}}$${\displaystyle N\times M}$ 矩阵${\displaystyle \ A}$满足${\displaystyle \ X=AZ+\mu }$.
• 存在${\displaystyle \mu }$和一个对称正定阵${\displaystyle \ \Sigma }$满足${\displaystyle \ X}$特征函数
${\displaystyle \phi _{X}\left(u;\mu ,\Sigma \right)=\exp \left(i\mu ^{T}u-{\frac {1}{2}}u^{T}\Sigma u\right)}$

${\displaystyle f_{\mathbf {x} }(x_{1},\ldots ,x_{k})={\frac {1}{\sqrt {(2\pi )^{k}|{\boldsymbol {\Sigma }}|}}}\exp \left(-{\frac {1}{2}}({\mathbf {x} }-{\boldsymbol {\mu }})^{\mathrm {T} }{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right),}$

{\displaystyle {\begin{aligned}f(x,y)&={\frac {1}{2\pi \sigma _{X}\sigma _{Y}{\sqrt {1-\rho ^{2}}}}}\exp \left(-{\frac {1}{2(1-\rho ^{2})}}\left[{\frac {(x-\mu _{X})^{2}}{\sigma _{X}^{2}}}+{\frac {(y-\mu _{Y})^{2}}{\sigma _{Y}^{2}}}-{\frac {2\rho (x-\mu _{X})(y-\mu _{Y})}{\sigma _{X}\sigma _{Y}}}\right]\right)\\\end{aligned}}}

${\displaystyle {\boldsymbol {\mu }}={\begin{pmatrix}\mu _{X}\\\mu _{Y}\end{pmatrix}},\quad {\boldsymbol {\Sigma }}={\begin{pmatrix}\sigma _{X}^{2}&\rho \sigma _{X}\sigma _{Y}\\\rho \sigma _{X}\sigma _{Y}&\sigma _{Y}^{2}\end{pmatrix}}.}$

## 參考資料

1. ^ UIUC, Lecture 21. The Multivariate Normal Distribution, 21.5:"Finding the Density".