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多变量正态分布亦称为多变量高斯分布。它是单维正态分布向多维的推广。它同矩阵正态分布有紧密的联系。
Quick Facts 记号, 参数 ...多元正态分布
概率密度函数 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).记号
N
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{\displaystyle {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}
参数
μ ∈ RN — 位置Σ ∈ RN×N — 协方差矩阵 (半正定)值域
x ∈ μ+span(Σ) ⊆ RN概率密度函数
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{\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）累积分布函数
解析表达式不存在期望值
μ众数
μ方差
Σ熵
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ln
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{\displaystyle {\frac {1}{2}}\ln((2\pi e)^{N}|{\boldsymbol {\Sigma }}|)}
矩生成函数
exp
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{\displaystyle \exp \!{\Big (}{\boldsymbol {\mu }}'\mathbf {t} +{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} {\Big )}}
特征函数
exp
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{\displaystyle \exp \!{\Big (}i{\boldsymbol {\mu }}'\mathbf {t} -{\tfrac {1}{2}}\mathbf {t} '{\boldsymbol {\Sigma }}\mathbf {t} {\Big )}}
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