Cov(X,Y)=E[(X−E[X])(Y−E[Y])]=E[XY]−E[X]E[Y] Cov(X,Y)=E\Big[(X-E[X])(Y-E[Y])\Big]=E[XY]-E[X]E[Y] Cov(X,Y)=E[(X−E[X])(Y−E[Y])]=E[XY]−E[X]E[Y]
Cov(X,Y)=E[XY]−E[X]E[Y]=∬R2xyf(x,y)dxdy−∫−∞∞xf(x)dx∫−∞∞yf(y)dy Cov(X,Y)=E[XY]-E[X]E[Y] =\iint_{R^{2}}xyf(x,y)dxdy-\int_{-\infty}^{\infty}xf(x)dx\int_{-\infty}^{\infty}yf(y)dyCov(X,Y)=E[XY]−E[X]E[Y]=∬R2xyf(x,y)dxdy−∫−∞∞xf(x)dx∫−∞∞yf(y)dy
Var[aX+bY+c]=a2Var[X]+b2Var[Y]+2abCov(X,Y) Var[aX+bY+c]=a^{2}Var[X]+b^{2}Var[Y]+2abCov(X,Y) Var[aX+bY+c]=a2Var[X]+b2Var[Y]+2abCov(X,Y)
可以看出常数项c的整体偏移并不影响方差和协方差的大小
协方差可看作方差的随机变量XXX 为不同变量时的推广
