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fix errors in section notes 07
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zlotus committed Sep 8, 2016
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"source": [
"# 多元高斯分布\n",
"\n",
"一个向量形式的随机变量$X=\\left[X_1\\cdots X_n\\right]^T$,期望为$\\mu\\in\\mathbb R^n$,协方差矩阵为$\\varSigma\\in\\mathbb S_{++}^n$(在线性代数中$\\mathbb S_{++}^n$为$n\\times n$正定对称矩阵空间,具体定义为$\\mathbb S_{++}^n=\\left\\{A\\in\\mathbb R^{n\\times n}: A=A^T,\\ \\forall x\\in\\mathbb R^n\\land x\\neq0\\right\\},\\ x^TAx\\gt0$),如果随机变量的概率密度函数(这篇笔记中我们使用$p(\\bullet)$表示概率密度函数,代替[概率论](sn02.ipynb)笔记中的$f_X(\\bullet)$)能够定义为:\n",
"一个向量形式的随机变量$X=\\left[X_1\\cdots X_n\\right]^T$,期望为$\\mu\\in\\mathbb R^n$,协方差矩阵为$\\varSigma\\in\\mathbb S_{++}^n$(在[线性代数](sn01.ipynb)笔记中$\\mathbb S_{++}^n$为$n\\times n$正定对称矩阵空间,具体定义为$\\mathbb S_{++}^n=\\left\\{A\\in\\mathbb R^{n\\times n}: A=A^T,\\ \\forall x\\in\\mathbb R^n\\land x\\neq0\\to x^TAx\\gt0\\right\\}$),如果随机变量的概率密度函数(这篇笔记中我们使用$p(\\bullet)$表示概率密度函数,代替[概率论](sn02.ipynb)笔记中的$f_X(\\bullet)$)能够定义为:\n",
"\n",
"$$\n",
"p\\left(x;\\mu,\\varSigma\\right)=\\frac{1}{\\left(2\\pi\\right)^{n/2}\\left\\lvert\\varSigma\\right\\rvert^{1/2}}\\exp\\left(-\\frac{1}{2}\\left(x-\\mu\\right)^T\\varSigma^{-1}\\left(x-\\mu\\right)\\right)\n",
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