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)=∑i=1n(Xi−X¯)(Yi−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]]

因为均值计算是线性的,即(a和b均为常数):
E[aX+bY]=aE[X]+bE[Y]
E[aX+bY]=aE[X]+bE[Y]

则我们有:
E[XY−E[X]Y−XE[Y]+E[X]E[Y]]
E[XY−E[X]Y−XE[Y]+E[X]E[Y]]

=E[XY]−E[X]E[Y]−E[X]E[Y]+E[X]E[Y]
=E[XY]−E[X]E[Y]−E[X]E[Y]+E[X]E[Y]

=E[XY]−E[X]E[Y]