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]