例 3.3.5 (n 维正态分布 (非退化情形))

设 $\mu ={\left({\mu }_1,{\mu }_2,\cdots ,{\mu }_n\right)}^{\prime }$, $\Sigma$ 为 $n$ 阶正定矩阵, 记 $\mathbf{X}={\left(x_1,x_2,\cdots ,x_n\right)}^{\prime }$ 若

$$f_{X_1,X_2,\cdots ,X_n}\left(x_1,x_2\cdots ,x_n\right)=\frac{1}{{\left(2\pi \right)}^{\frac{n}{2}}{\left|\Sigma \right|}^{\frac{n}{2}}}\exp \left\{-\frac{1}{2}{\left(\mathbf{X}-\mu \right)}^{\prime }{\Sigma }^{-1}\left(\mathbf{X}-\mu \right)\right\}.$$

则称 ${\left(X_1,X_2,\cdots ,X_n\right)}^{\prime }$ 服从 $n$ 维正态分布, 记作 $\mathbf{X}\sim N\left(\mu ,\Sigma \right)$.

不难看出, 在 例 3.3.4 (二维正态分布) 中,

$$\mu =\left(\begin{array}{l}{\mu }_1\\ {\mu }_2\end{array}\right),\Sigma =\left(\begin{array}{ll}{\sigma }_1^2& \rho {\sigma }_1{\sigma }_2\\ \rho {\sigma }_1{\sigma }_2& {\sigma }_2^2\end{array}\right).$$