高斯分布
定义式
\[
f(x)={\frac{1}{{\sqrt{2\pi}}\,\sigma}}e^{-\,{\frac{(x-\mu)^{2}}{2\sigma^{2}}}}\quad-\infty<x<+\infty
\]
记为 \(X\sim N\left(\mu,\sigma^{2}\right)\)
特征函数
\[
Q_{X}(u)=\exp[j u m-\frac{u^{2}\sigma^{2}}{2}],\quad-\infty<u<+\infty
\]
标准高斯分布
当 \(\mu=0\)\(\sigma=1\) 时，称为标准高斯分布，记作\(X\sim N(0,1)\)
\[
f_X(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}},\quad-\infty<x<+\infty
\]
特征函数
\[
Q_{X}(u)=\exp[-\frac{u^{2}}{2}],\quad -\infty<u<+\infty
\]
\(\text{可得积分}\int_{-\infty}^{+\infty}e^{-\frac{t^2}{2}}dt=\sqrt{2\pi}\text{ 或}\int_{-\infty}^{+\infty}e^{-t^2}dt=\sqrt{\pi}\)
性质
- \(P(\mid X\mid<a)=2\Phi(a)-1\)
- \(X\sim N(\mu,\sigma^{2}),Y=k X+b, 则 Y\sim N(k\mu+b,k^{2}\sigma^{2})\)
标准化
\[
Y=\frac{X-\mu}{\sigma}\sim N(0,1)
\]
例 设 \(X\sim N(1,4)\) , 求 \(P(0\leq X\leq1.6)\)
解 \(P(0\leq X\leq1.6)=\Phi\bigg(\frac{1.6-1}{2}\bigg)-\Phi\bigg(\frac{0-1}{2}\bigg)\) = Φ(0.3) − Φ(−0.5)
高斯变量的 n 阶矩
标准高斯 n 阶矩
\[
X\sim N(0,1),E[X^n]=\begin{cases}0,&n\text{为奇数}\\1\cdot3\cdot5\cdots(n-1)=(n-1)!,&n\text{为偶数}&\end{cases}
\]
一般高斯变量 Y 的 n 阶中心矩：
\(\begin{aligned}&\text{当}Y\sim N(m_\gamma,\sigma^2)\text{时},\because X=\frac{Y-m_\gamma}{\sigma},\\&E[X^n]=E[(\frac{Y-m_Y}{\sigma})^n]=\frac{E[(Y-m_Y)^n]}{\sigma^n},\\&\therefore E[(Y-m_\gamma)^n]=\sigma^nE[X^n]=\begin{cases}0,&n\text{为奇数}\\\sigma^n(n-1)!,&n\text{为偶数}&&\end{cases}\end{aligned}\)
n 维高斯分布
可记为 \(\pmb{X}\sim N(\pmb{M}_{X},\pmb{{C}}_{X})\)
随机变量与均值为
\[
\pmb{X}=\left[\begin{array}{c}{\pmb{X}_{1}}\\ {\vdots}\\ {\pmb{X}_{n}}\end{array}\right],\quad\pmb{M}_{X}=\left[\begin{array}{c}{\pmb{m}_{1}}\\ {\vdots}\\ {\pmb{m}_{n}}\end{array}\right]
\]
方差为
\[
{{D}}X=C_{X}=E{\Big[}\left(X-M_{X}\right)\left(X-M_{X}\right)^{\mathrm{T}}{\Big]}={\left[\begin{array}{l l l l}{C_{11}}&{C_{12}}&{\cdots}&{C_{1n}}\\ {C_{21}}&{C_{22}}&{\cdots}&{C_{2n}}\\ {\vdots}&{\vdots}&&{\vdots}\\ {C_{n1}}&{C_{n2}}&{\cdots}&{C_{n n}}\end{array}\right]}_{n\times n}
\]
其中对角线为方差，其余为协方差
联合概率密度 有以下矩阵形式
\[
f_{X}\left(x_{1},\cdots,x_{n}\right)={\frac{1}{\left(2\pi\right)^{\frac{n}{2}}\left|C_{X}\right|^{\frac{1}{2}}}}\exp\left[-{\frac{\left(X-M_{X}\right)^{\mathrm{T}}C_{X}^{-1}\left(X-M_{X}\right)}{2}}\right]
\]
同时，特征函数有
\[
Q_{X}\left(u_{1},\cdots,u_{n}\right)=E\bigg[\mathrm{e}^{\mathrm{i}U^{\mathrm{T}}X}\bigg]=\exp\bigg(\mathrm{j}M_{X}^{\mathrm{T}}U-\frac{U^{\mathrm{T}}C_{X}U}{2}\bigg)
\]
n 维高斯变量的特点
- 不相关(线性不相关) \(=\) 独立
证明
下面用独立的定义（联合概率密度函数为边缘概率密度函数之积）证明，亦可用特征函数
∵ 互不相关
∴ 方差矩阵变为对角阵
\[
\mathbf{C}_{X}=\left(\begin{array}{c c c}{\sigma_{1}^{2}}&{\cdots}&{C_{1n}}\\ {\vdots}&{\ddots}&{\vdots}\\ {C_{n1}}&{\cdots}&{\sigma_{n}^{2}}\end{array}\right)=\left(\begin{array}{c c c}{\sigma_{1}^{2}}&{\cdots}&{0}\\ {\vdots}&{\ddots}&{\vdots}\\ {0}&{\cdots}&{\sigma_{n}^{2}}\end{array}\right)
\]
已知 n 维联合概率密度函数为
\[
f_{X}\left(x_{1},\cdots,x_{n}\right)={\frac{1}{\left(2\pi\right)^{\frac{n}{2}}\left|C_{X}\right|^{\frac{1}{2}}}}\exp\left[-{\frac{\left(X-M_{X}\right)^{\mathrm{T}}C_{X}^{-1}\left(X-M_{X}\right)}{2}}\right]
\]
其中
\[
\pmb{X}-\pmb{M}_{X}=\left[\begin{array}{c}{X_{1}-m_{1}}\\ {\vdots}\\ {X_{n}-m_{n}}\end{array}\right],\:\:(\pmb{X}-\pmb{M}_{X})^{T}=\left[X_{1}-m_{1},\ldots,X_{n}-m_{n}\right]
\]
\[
\mathbf{C}_{X}{}^{-1}=\left(\begin{array}{c c c}{\frac{1}{\sigma_{1}^{2}}}&{\cdots}&{0}\\ {\vdots}&{\ddots}&{\vdots}\\ {0}&{\cdots}&{\frac{1}{\sigma_{n}^{2}}}\end{array}\right),|\mathbf{C}_{X}|^{\frac{1}{2}}=\prod_{i=1}^{n}\sigma_{i}
\]
故可整理得到联合概率密度函数
\[
{\begin{array}{c}{f_{X}\left(x_{1},\cdots,x_{n}\right)={\displaystyle{\frac{1}{\left({\sqrt{2\pi}}\right)^{n}\prod_{i=1}^{n}\sigma_{i}}}\exp\left[{-{\frac{\sum_{i=0}^{n}\left({\frac{x_{i}-m_{i}}{\sigma_{i}}}\right)^{2}}{2}}}\right]}=}\\ {\displaystyle\prod_{i=1}^{n}{\frac{1}{\left({\sqrt{2\pi}}\right)^{n}\sigma_{i}}}\exp\left[{-{\frac{\left({\frac{x_{i}-m_{i}}{\sigma_{i}}}\right)^{2}}{2}}}\right]}\\ {=\displaystyle\prod_{i=1}^{n}f_{X_{i}}(x)}\end{array}}
\]
- n 维高斯变量经过线性变换后仍然服从高斯分布
- 高斯矢量的边缘分布仍是高斯分布