多元高斯分布函数

1、$n$元向量

假设$n$元随机变量$X$
$$\begin{array}{rl}X& =[X_1,X_2,\cdots ,X_i,\cdots ,X_n{]}^T\\ \mu & =[{\mu }_1,{\mu }_2,\cdots ,{\mu }_i,\cdots ,{\mu }_n{]}^T\\ \sigma & =[{\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_i,\cdots ,{\sigma }_n{]}^T\\ X_i& \sim N({\mu }_i,{\sigma }_i^2)\end{array}$$
$\Sigma$为协方差矩阵。
$$\begin{array}{rl}\Sigma & =\left[\begin{array}{cccc}Conv(X_1,X_1)& Conv(X_1,X_2)& \cdots & Conv(X_1,X_n)\\ Conv(X_2,X_1)& Conv(X_2,X_2)& \cdots & Conv(X_2,X_n)\\ ⋮& ⋮& \ddots & ⋮\\ Conv(X_n,X_1)& Conv(X_n,X_2)& \cdots & Conv(X_n,X_n)\end{array}\right]\end{array}$$
当$X_1,X_2,\cdots ,X_i,\cdots ,X_n$之间相互独立时，有
$$\begin{array}{rl}\Sigma & =\left[\begin{array}{cccc}{\sigma }_1^2& 0& \cdots & 0\\ 0& {\sigma }_2^2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\sigma }_n^2\end{array}\right]\end{array}$$

2 、$n$元高斯分布

$$\begin{array}{rl}p(X)& =\frac{1}{(2\pi {)}^{\frac{n}{2}}\cdot |\Sigma {|}^{\frac{1}{2}}}\cdot e^{-\frac{(X-\mu {)}^T{\Sigma }^{-1}(X-\mu )}{2}}\end{array}$$
其中，$|\Sigma |$为协方差矩阵$\Sigma$的行列式

3、1元高斯分布

此时
$$\begin{array}{rl}X& =[X_1]\\ \mu & =[{\mu }_1]\\ \sigma & =[{\sigma }_1]\\ X_1& \sim N({\mu }_1,{\sigma }_1^2)\\ \Sigma & =[{\sigma }_1^2]\end{array}$$

$$\begin{array}{rl}p(X_1)& =\frac{1}{(2\pi {)}^{\frac{n}{2}}\cdot |\Sigma {|}^{\frac{1}{2}}}\cdot e^{-\frac{(X-\mu {)}^T{\Sigma }^{-1}(X-\mu )}{2}}\\ & =\frac{1}{(2\pi {)}^{\frac{1}{2}}\cdot ({\sigma }_1^2{)}^{\frac{1}{2}}}\cdot e^{-\frac{(X_1-{\mu }_1{)}^T({\sigma }_1^2{)}^{-1}(X_1-{\mu }_1)}{2}}\\ & =\frac{1}{\sqrt{2\pi }\cdot {\sigma }_1}\cdot e^{-\frac{(X_1-{\mu }_1{)}^2}{2\cdot {\sigma }_1^2}}\end{array}$$

2、相互独立的2元高斯分布

此时
$$\begin{array}{rl}X& =[X_1,X_2{]}^T\\ \mu & =[{\mu }_1,{\mu }_2{]}^T\\ \sigma & =[{\sigma }_1,{\sigma }_2{]}^T\\ X_i& \sim N({\mu }_i,{\sigma }_i^2)\\ \Sigma & =\left[\begin{array}{cc}{\sigma }_1^2& 0\\ 0& {\sigma }_2^2\end{array}\right]\end{array}$$
$$\begin{array}{rl}p(X)& =p([X_1,X_2{]}^T)\\ & =\frac{1}{(2\pi {)}^{\frac{n}{2}}\cdot |\Sigma {|}^{\frac{1}{2}}}\cdot e^{-\frac{(X-\mu {)}^T{\Sigma }^{-1}(X-\mu )}{2}}\\ & =\frac{1}{(2\pi {)}^{\frac{2}{2}}\cdot {\left|\begin{array}{cc}{\sigma }_1^2& 0\\ 0& {\sigma }_2^2\end{array}\right|}^{\frac{1}{2}}}\cdot e^{-\frac{(\left[\begin{array}{c}{X}_1\\ X_2\end{array}\right]-\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right]{)}^T{\left[\begin{array}{cc}{\sigma }_1^2& 0\\ 0& {\sigma }_2^2\end{array}\right]}^{-1}(\left[\begin{array}{c}{X}_1\\ X_2\end{array}\right]-\left[\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\right])}{2}}\\ & =\frac{1}{2\pi \cdot {\sigma }_1\cdot {\sigma }_2}\cdot e^{-\frac{{\left[\begin{array}{c}{X}_1-{\mu }_1\\ X_2-{\mu }_2\end{array}\right]}^T\left[\begin{array}{cc}\frac{1}{{\sigma }_1^2}& 0\\ 0& \frac{1}{{\sigma }_2^2}\end{array}\right]\left[\begin{array}{c}{X}_1-{\mu }_1\\ X_2-{\mu }_2\end{array}\right]}{2}}\\ & =\frac{1}{2\pi \cdot {\sigma }_1\cdot {\sigma }_2}\cdot e^{-\frac{\left[\begin{array}{c}{X}_1-{\mu }_1,X_2-{\mu }_2\end{array}\right]\left[\begin{array}{cc}\frac{1}{{\sigma }_1^2}& 0\\ 0& \frac{1}{{\sigma }_2^2}\end{array}\right]\left[\begin{array}{c}{X}_1-{\mu }_1\\ X_2-{\mu }_2\end{array}\right]}{2}}\\ & =\frac{1}{2\pi \cdot {\sigma }_1\cdot {\sigma }_2}\cdot e^{-\frac{\left[\begin{array}{c}\frac{X_1-{\mu }_1}{{\sigma }_1^2},\frac{X_2-{\mu }_2}{{\sigma }_2^2}\end{array}\right]\left[\begin{array}{c}{X}_1-{\mu }_1\\ X_2-{\mu }_2\end{array}\right]}{2}}\\ & =\frac{1}{2\pi \cdot {\sigma }_1\cdot {\sigma }_2}\cdot e^{-\frac{\frac{(X_1-{\mu }_1{)}^2}{{\sigma }_1^2}-\frac{(X_2-{\mu }_2{)}^2}{{\sigma }_2^2}}{2}}\\ & =\frac{1}{\sqrt{2\pi }\cdot {\sigma }_1}\cdot e^{-\frac{(X_1-{\mu }_1{)}^2}{2{\sigma }_1^2}}\cdot \frac{1}{\sqrt{2\pi }\cdot {\sigma }_2}\cdot e^{-\frac{(X_2-{\mu }_2{)}^2}{2{\sigma }_2^2}}\end{array}$$