- 若将\(X\sim N_p(\mu,\Sigma)\)进行分割：
\[ X= \left[ \begin{array}{c} X^{(1)}_r\\ X^{(2)}_{p-r} \end{array} \right]， \mu= \left[ \begin{array}{c} \mu^{(1)}_r\\ \mu^{(2)}_{p-r} \end{array} \right]， \Sigma= \left[ \begin{array}{c|c} \Sigma_{11} &\Sigma_{12}\\ \hline \Sigma_{21} &\Sigma_{22} \end{array} \right]>0，(\Sigma_{11}为r\times r方阵) \]
一、独立性
设 \(p\) 维随机向量 \(X\sim N_p(\mu,\Sigma)\),
\[ X= \left[ \begin{array}{c} X^{(1)}\\ X^{(2)} \end{array} \right]\sim \left( \left[ \begin{array}{c} \mu^{(1)}\\ \mu^{(2)} \end{array} \right]， \left[ \begin{array}{cc} \Sigma_{11} &\Sigma_{12}\\ \Sigma_{21} &\Sigma_{22} \end{array} \right] \right) \]
则
\[ X^{(1)}与 X^{(2)}相互独立\ \leftrightarrows\ \Sigma_{12}=O \]
(证明)
设\(\Sigma_{12}=O\)，则\(X\)的联合密度函数为：
\[ \begin{align} f(x^{(1)},x^{(2)})=& \frac1{(2\pi)^{p/2}|\Sigma|^{1/2}}exp\left(-\frac12(x-\mu)' \left[ \begin{array}{cc} \Sigma_{11}&O\\ O&\Sigma_{22} \end{array} \right]^{-1} (x-\mu) \right)\\ =& \frac1{(2\pi)^{r/2}|\Sigma_{11}|^{1/2}}exp\left(-\frac12(x^{(1)}-\mu^{(1)})' \Sigma_{11}^{-1} (x^{(1)}-\mu^{(1)}) \right)\\ &\cdot \frac1{(2\pi)^{(p-r)/2}|\Sigma_{22}|^{1/2}}exp\left(-\frac12(x^{(2)}-\mu^{(2)})' \Sigma_{22}^{-1} (x^{(2)}-\mu^{(2)}) \right)\\ =&f_1(x^{(1)})\cdot f_2(x^{(2)}) \end{align} \]因此\(X^{(1)},X^{(2)}\)相互独立。
- 设\(r_i\geq1,(i=1,\dots,k)\),且\(r_1+r_2+\dots+r_k=p\),则有
\[ X= \left[ \begin{array}{c} X^{(1)}\\ \vdots\\ X^{(k)} \end{array} \right]\sim N_p \left( \left[ \begin{array}{c} \mu^{(1)}\\ \vdots\\ \mu^{(k)} \end{array} \right]， \left[ \begin{array}{ccc} \Sigma_{11} &\cdots &\Sigma_{1k}\\ \vdots&&\vdots\\ \Sigma_{k1} &\cdots &\Sigma_{kk} \end{array} \right]_{p\times p} \right) \]
则\(X^{(1)},X^{(2)},\dots,X^{(k)}\)相互独立 \(\leftrightarrows\) \(\Sigma_{ij}=O,(i\neq j)\).
- 设\(X=(X_1,\dots,X_p)'\sim N_p(\mu,\Sigma)\),若\(\Sigma\)为对角矩阵，则\(X_1,\dots,X_p\)相互独立。
二、条件分布
当\(X_2\)给定时，\(X_1\)的条件密度为：
\[ f_1(x_1|x_2)=\frac{f(x_1,x_2)}{f_2(x_2)} \]
\[ \begin{align} f(x_1,x_2)= &=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}exp\left\{-\frac{1}{2(1-\rho^2)}[(\frac{x_1-\mu_1}{\sigma_1})^2-2\rho(\frac{x_1-\mu_1}{\sigma_1})(\frac{x_2-\mu_2}{\sigma_2})+(\frac{x_2-\mu_2}{\sigma_2})^2]\right\}\\ &=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}exp\left\{-\frac{1}{2(1-\rho^2)}[(\frac{x_1-\mu_1}{\sigma_1})^2-2\rho(\frac{x_1-\mu_1}{\sigma_1})(\frac{x_2-\mu_2}{\sigma_2})+(1-\rho^2)(\frac{x_2-\mu_2}{\sigma_2})^2+\rho^2(\frac{x_2-\mu_2}{\sigma_2})^2] \right\}\\ &=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}exp\left\{-\frac{1}{2}(\frac{x_2-\mu_2}{\sigma_2})^2\right\}\cdot exp\left\{-\frac{1}{2(1-\rho^2)}[(\frac{x_1-\mu_1}{\sigma_1})^2-2\rho(\frac{x_1-\mu_1}{\sigma_1})(\frac{x_2-\mu_2}{\sigma_2})+\rho^2(\frac{x_2-\mu_2}{\sigma_2})^2] \right\}\\ &=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}exp\left\{-\frac{1}{2}(\frac{x_2-\mu_2}{\sigma_2})^2\right\}\cdot exp\left\{-\frac{1}{2(1-\rho^2)}[(\frac{x_1-\mu_1}{\sigma_1})-\rho(\frac{x_2-\mu_2}{\sigma_2})]^2\right\}\\ &=\frac{1}{\sqrt{2\pi}\sigma_2}exp\left\{-\frac{1}{2}(\frac{x_2-\mu_2}{\sigma_2})^2\right\}\cdot\frac{1}{\sqrt{2\pi}\sigma_1\sqrt{1-\rho^2}}\cdot exp\left\{-\frac{1}{2(1-\rho^2)\sigma_1^2}[x_1-\mu_1-\rho\frac{\sigma_1}{\sigma_2}(x_2-\mu_2)]^2\right\}\\ &=f_2(x_2)\cdot f(x_1|x_2) \end{align} \]
其中
\[ f(x_1|x_2)=\frac{1}{\sqrt{2\pi}\sigma_1\sqrt{1-\rho^2}}\cdot exp\left\{ -\frac{1}{2(1-\rho^2)\sigma_1^2}[x_1-\left(\mu_1 +\rho\frac{\sigma_1}{\sigma_2}(x_2-\mu_2)\right)]^2 \right\}\\ \]
由定义：
\[ (X_1|X_2)\sim N_1\left(\mu_1+\rho\frac{\sigma_1}{\sigma_2}(x_2-\mu_2)，\sigma^2(1-\rho^2)\right) \]
将其推广到多维：
设
\[ X= \left[ \begin{array}{c} X^{(1)}_r\\ X^{(2)}_{p-r} \end{array} \right]\sim N_p(\mu,\Sigma)，(\Sigma>0) \]
则当\(X^{(2)}\)给定时，\(X^{(1)}\)的条件分布为：
\[ (X^{(1)}|X^{(2)})\sim N_r(\mu_{1\cdot2}，\Sigma_{11\cdot2}) \]
其中
\[ \mu_{1\cdot2}=\mu^{(1)}+\Sigma_{12}\Sigma_{22}^{-1}(x^{(2)}-\mu^{(2)})\\ \Sigma_{11\cdot2}=\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21} \]