9.4. 期望向量与协方差矩阵#
- 数学期望向量
记 \(n\) 维随机向量 \(\boldsymbol{X}=\left(X_{1}, X_{2}, \cdots, X_{n}\right)'\) ，若其每个分量的数学期望都存在，则称
\[E(\boldsymbol{X})=\left(E\left(X_{1}\right), E\left(X_{2}\right), \cdots, E\left(X_{n}\right)\right)'\]
为 \(n\) 维随机向量的数学期望向量。
- 协方差矩阵
而称
\[\begin{split}
E(\boldsymbol{X}-E(\boldsymbol{X}))(\boldsymbol{X}-E(\boldsymbol{X}))'
=\left(\begin{matrix}
\text{Var}\left(X_{1}\right) & \text{Cov}\left(X_{1}, X_{2}\right) & \cdots & \text{Cov}\left(X_{1}, X_{n}\right) \\
\text{Cov}\left(X_{2}, X_{1}\right) & \text{Var}\left(X_{2}\right) & \cdots & \text{Cov}\left(X_{2}, X_{n}\right) \\
\vdots & \vdots & & \vdots \\
\text{Cov}\left(X_{n}, X_{1}\right) & \text{Cov}\left(X_{n}, X_{2}\right) & \cdots & \text{Var}\left(X_{n}\right)
\end{matrix}\right)
\end{split}\]
为该随机向量的方差—协方差矩阵，记为 \(\text{Cov}(\boldsymbol{X})\)
Theorem 9.4
\(n\) 维随机向量的协方差矩阵 \(\text{Cov}(\boldsymbol{X})=\left\{\text{Cov}\left(X_{i}, X_{j}\right)\right\}_{n\times n}\) 是一个对称非负定矩阵。