多变量分布矩



  • 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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