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    多变量分布矩

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    • 李东岳
      李东岳 管理员 last edited by 李东岳

      二变量高斯分布:
      \begin{equation}
      f(u,v)=\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}\exp\left(-\frac{1}{2(1-\rho^2)}\left[\frac{(u-\mu_1)^2}{\sigma_1^2}-\frac{2\rho(u-\mu_1)(v-\mu_2)}{\sigma_1\sigma_2}+\frac{(v-\mu_2)^2}{\sigma_2^2}\right]\right)
      \end{equation}
      MGF为:
      \begin{equation}
      m_{i,j}=\exp\left(i\mu_1+j\mu_2+0.5(\sigma_1^2i^2+2\rho\sigma_1\sigma_2ij+\sigma_2^2j^2)\right)
      \end{equation}
      纯矩计算方法为
      \begin{equation}
      \begin{split}
      m_{0,0}&=1\\
      m_{1,0}&=\mu\\
      m_{2,0}&=\mu^2+\sigma^2\\
      m_{3,0}&=\mu^3+3\mu\sigma^2\\
      \end{split}
      \end{equation}
      假设$\mu_1=10,\mu_2=20,\sigma_1=\sigma_2=2,\rho=0.5$,有纯矩:
      \begin{split}
      m_{0,0}&=1\\
      m_{1,0}&=10\\
      m_{2,0}&=104\\
      m_{3,0}&=1120\\
      m_{0,1}&=20\\
      m_{0,2}&=404\\
      m_{0,3}&=8240\\
      \end{split}
      同时有混合矩
      \begin{equation}
      m_{i,j}=\exp\left(36\right)
      \end{equation}
      不行,混合矩计算方法不对

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