协方差公式推导
协方差公式推导cov(X,Y)=∑ni=1(Xi−X¯)(Yi−Y¯)n=E[(X−E[X])(Y−E[Y])] cov(X,Y)=\frac{\sum_{i=1}^{n}(X_i-\bar{X})(Y_i-\bar{Y})}{n}=E[(X-E[X])(Y-E[Y])]=E[XY−E[X]Y−XE[Y]+E[X]E[Y]] =E[XY-E[X]Y-XE[Y]+E[X]E[Y]]因为均值
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协方差公式推导
cov(X,Y)=∑ni=1(Xi−X¯)(Yi−Y¯)n=E[(X−E[X])(Y−E[Y])]
<script type="math/tex; mode=display" id="MathJax-Element-1"> cov(X,Y)=\frac{\sum_{i=1}^{n}(X_i-\bar{X})(Y_i-\bar{Y})}{n}=E[(X-E[X])(Y-E[Y])] </script>
=E[XY−E[X]Y−XE[Y]+E[X]E[Y]]
<script type="math/tex; mode=display" id="MathJax-Element-2"> =E[XY-E[X]Y-XE[Y]+E[X]E[Y]] </script>
因为均值计算是线性的，即（a和b均为常数）：
E[aX+bY]=aE[X]+bE[Y]
<script type="math/tex; mode=display" id="MathJax-Element-3"> E[aX+bY]=aE[X]+bE[Y] </script>
则我们有：
E[XY−E[X]Y−XE[Y]+E[X]E[Y]]
<script type="math/tex; mode=display" id="MathJax-Element-4"> E[XY-E[X]Y-XE[Y]+E[X]E[Y]] </script>
=E[XY]−E[X]E[Y]−E[X]E[Y]+E[X]E[Y]
<script type="math/tex; mode=display" id="MathJax-Element-5"> =E[XY]-E[X]E[Y]-E[X]E[Y]+E[X]E[Y] </script>
=E[XY]−E[X]E[Y]
<script type="math/tex; mode=display" id="MathJax-Element-6"> =E[XY]-E[X]E[Y] </script>
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