1、 n n n元向量
假设 n n n元随机变量 X X X
X = X 1 , X 2 , ⋯ , X i , ⋯ , X n T μ = μ 1 , μ 2 , ⋯ , μ i , ⋯ , μ n T σ = σ 1 , σ 2 , ⋯ , σ i , ⋯ , σ n T X i ∼ N ( μ i , σ i 2 ) \begin{split} 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{split} XμσXi=X1,X2,⋯,Xi,⋯,XnT=μ1,μ2,⋯,μi,⋯,μnT=σ1,σ2,⋯,σi,⋯,σnT∼N(μi,σi2)
Σ \Sigma Σ为协方差矩阵。
Σ = C o n v ( X 1 , X 1 ) C o n v ( X 1 , X 2 ) ⋯ C o n v ( X 1 , X n ) C o n v ( X 2 , X 1 ) C o n v ( X 2 , X 2 ) ⋯ C o n v ( X 2 , X n ) ⋮ ⋮ ⋱ ⋮ C o n v ( X n , X 1 ) C o n v ( X n , X 2 ) ⋯ C o n v ( X n , X n ) \begin{split} \Sigma&=\left\\begin{matrix} 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) \\\\ \\vdots \& \\vdots \& \\ddots \& \\vdots \\\\ Conv(X_n,X_1) \& Conv(X_n,X_2) \& \\cdots \& Conv(X_n,X_n) \\\\ \\end{matrix}\\right \end{split} Σ= Conv(X1,X1)Conv(X2,X1)⋮Conv(Xn,X1)Conv(X1,X2)Conv(X2,X2)⋮Conv(Xn,X2)⋯⋯⋱⋯Conv(X1,Xn)Conv(X2,Xn)⋮Conv(Xn,Xn)
当 X 1 , X 2 , ⋯ , X i , ⋯ , X n X_1,X_2,\cdots,X_i,\cdots ,X_n X1,X2,⋯,Xi,⋯,Xn之间相互独立时，有
Σ = σ 1 2 0 ⋯ 0 0 σ 2 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ σ n 2 \begin{split} \Sigma&=\left\\begin{matrix} \\sigma_1\^2 \& 0 \& \\cdots \& 0 \\\\ 0 \& \\sigma_2\^2 \& \\cdots \& 0 \\\\ \\vdots \& \\vdots \& \\ddots \& \\vdots \\\\ 0 \& 0 \& \\cdots \& \\sigma_n\^2 \\\\ \\end{matrix}\\right \end{split} Σ= σ120⋮00σ22⋮0⋯⋯⋱⋯00⋮σn2
2 、 n n n元高斯分布
p ( X ) = 1 ( 2 π ) n 2 ⋅ ∣ Σ ∣ 1 2 ⋅ e − ( X − μ ) T Σ − 1 ( X − μ ) 2 \begin{split} 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{split} p(X)=(2π)2n⋅∣Σ∣211⋅e−2(X−μ)TΣ−1(X−μ)
其中， ∣ Σ ∣ |\Sigma| ∣Σ∣为协方差矩阵 Σ \Sigma Σ的行列式
3、1元高斯分布
此时
X = X 1 μ = μ 1 σ = σ 1 X 1 ∼ N ( μ 1 , σ 1 2 ) Σ = σ 1 2 \begin{split} X&=X_1\\ \mu&= \\mu_1\\ \sigma&= \\sigma_1\\ X_1&\sim N(\mu_1,\sigma_1^2)\\ \Sigma&=\\sigma_1\^2 \end{split} XμσX1Σ=X1=μ1=σ1∼N(μ1,σ12)=σ12
p ( X 1 ) = 1 ( 2 π ) n 2 ⋅ ∣ Σ ∣ 1 2 ⋅ e − ( X − μ ) T Σ − 1 ( X − μ ) 2 = 1 ( 2 π ) 1 2 ⋅ ( σ 1 2 ) 1 2 ⋅ e − ( X 1 − μ 1 ) T ( σ 1 2 ) − 1 ( X 1 − μ 1 ) 2 = 1 2 π ⋅ σ 1 ⋅ e − ( X 1 − μ 1 ) 2 2 ⋅ σ 1 2 \begin{split} 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{split} p(X1)=(2π)2n⋅∣Σ∣211⋅e−2(X−μ)TΣ−1(X−μ)=(2π)21⋅(σ12)211⋅e−2(X1−μ1)T(σ12)−1(X1−μ1)=2π ⋅σ11⋅e−2⋅σ12(X1−μ1)2
2、相互独立的2元高斯分布
此时
X = X 1 , X 2 T μ = μ 1 , μ 2 T σ = σ 1 , σ 2 T X i ∼ N ( μ i , σ i 2 ) Σ = σ 1 2 0 0 σ 2 2 \begin{split} 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{matrix} \\sigma_1\^2 \& 0 \\\\ 0 \& \\sigma_2\^2 \\\\ \\end{matrix}\\right \end{split} XμσXiΣ=X1,X2T=μ1,μ2T=σ1,σ2T∼N(μi,σi2)=σ1200σ22
p ( X ) = p ( X 1 , X 2 T ) = 1 ( 2 π ) n 2 ⋅ ∣ Σ ∣ 1 2 ⋅ e − ( X − μ ) T Σ − 1 ( X − μ ) 2 = 1 ( 2 π ) 2 2 ⋅ ∣ σ 1 2 0 0 σ 2 2 ∣ 1 2 ⋅ e − ( X 1 X 2 − μ 1 μ 2 ) T σ 1 2 0 0 σ 2 2 − 1 ( X 1 X 2 − μ 1 μ 2 ) 2 = 1 2 π ⋅ σ 1 ⋅ σ 2 ⋅ e − X 1 − μ 1 X 2 − μ 2 T 1 σ 1 2 0 0 1 σ 2 2 X 1 − μ 1 X 2 − μ 2 2 = 1 2 π ⋅ σ 1 ⋅ σ 2 ⋅ e − X 1 − μ 1 , X 2 − μ 2 1 σ 1 2 0 0 1 σ 2 2 X 1 − μ 1 X 2 − μ 2 2 = 1 2 π ⋅ σ 1 ⋅ σ 2 ⋅ e − X 1 − μ 1 σ 1 2 , X 2 − μ 2 σ 2 2 X 1 − μ 1 X 2 − μ 2 2 = 1 2 π ⋅ σ 1 ⋅ σ 2 ⋅ e − ( X 1 − μ 1 ) 2 σ 1 2 − ( X 2 − μ 2 ) 2 σ 2 2 2 = 1 2 π ⋅ σ 1 ⋅ e − ( X 1 − μ 1 ) 2 2 σ 1 2 ⋅ 1 2 π ⋅ σ 2 ⋅ e − ( X 2 − μ 2 ) 2 2 σ 2 2 \begin{split} 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{matrix} \sigma_1^2 & 0 \\ 0 & \sigma_2^2 \\ \end{matrix}\right|^{\frac{1}{2}}}\cdot e^{-\frac{\Bigg(\left\\begin{matrix} X_1 \\\\ X_2 \\\\ \\end{matrix}\\right-\left\\begin{matrix} \\mu_1 \\\\ \\mu_2 \\\\ \\end{matrix}\\right\Bigg)^T\left\\begin{matrix} \\sigma_1\^2 \& 0 \\\\ 0 \& \\sigma_2\^2 \\\\ \\end{matrix}\\right^{-1}\Bigg(\left\\begin{matrix} X_1 \\\\ X_2 \\\\ \\end{matrix}\\right-\left\\begin{matrix} \\mu_1 \\\\ \\mu_2 \\\\ \\end{matrix}\\right\Bigg)}{2}} \\ &=\frac{1}{2\pi\cdot \sigma_1\cdot \sigma_2}\cdot e^{-\frac{\left\\begin{matrix} X_1 -\\mu_1\\\\ X_2 -\\mu_2 \\\\ \\end{matrix}\\right^T\left\\begin{matrix} \\frac{1}{\\sigma_1\^2} \& 0 \\\\ 0 \& \\frac{1}{\\sigma_2\^2} \\\\ \\end{matrix}\\right \left\\begin{matrix} X_1-\\mu_1 \\\\ X_2-\\mu_2 \\\\ \\end{matrix}\\right}{2}} \\ &=\frac{1}{2\pi\cdot \sigma_1\cdot \sigma_2}\cdot e^{-\frac{\left\\begin{matrix} X_1 -\\mu_1, X_2 -\\mu_2 \\\\ \\end{matrix}\\right\left\\begin{matrix} \\frac{1}{\\sigma_1\^2} \& 0 \\\\ 0 \& \\frac{1}{\\sigma_2\^2} \\\\ \\end{matrix}\\right \left\\begin{matrix} X_1-\\mu_1 \\\\ X_2-\\mu_2 \\\\ \\end{matrix}\\right}{2}} \\ &=\frac{1}{2\pi\cdot \sigma_1\cdot \sigma_2}\cdot e^{-\frac{\left\\begin{matrix} \\frac{X_1 -\\mu_1}{\\sigma_1\^2}, \\frac{X_2 -\\mu_2}{\\sigma_2\^2} \\\\ \\end{matrix}\\right \left\\begin{matrix} X_1-\\mu_1 \\\\ X_2-\\mu_2 \\\\ \\end{matrix}\\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{split} p(X)=p(X1,X2T)=(2π)2n⋅∣Σ∣211⋅e−2(X−μ)TΣ−1(X−μ)=(2π)22⋅ σ1200σ22 211⋅e−2(X1X2−μ1μ2)Tσ1200σ22−1(X1X2−μ1μ2)=2π⋅σ1⋅σ21⋅e−2X1−μ1X2−μ2Tσ12100σ221X1−μ1X2−μ2=2π⋅σ1⋅σ21⋅e−2X1−μ1,X2−μ2σ12100σ221X1−μ1X2−μ2=2π⋅σ1⋅σ21⋅e−2σ12X1−μ1,σ22X2−μ2X1−μ1X2−μ2=2π⋅σ1⋅σ21⋅e−2σ12(X1−μ1)2−σ22(X2−μ2)2=2π ⋅σ11⋅e−2σ12(X1−μ1)2⋅2π ⋅σ21⋅e−2σ22(X2−μ2)2