协方差cov(x,-y)= -cov(x,y)吗

题目

协方差cov(x,-y)= -cov(x,y)吗

答案

设：E{X}=a,E{Y}=b
则：cov(x,y) = E{(X-a)(Y-b)}
= E{XY}-ab-ab+ab
= E{XY}-ab
所以：cov(x,-y) = E{(X-a)(-Y+b)}
= -E{XY}+ab
= -cov(X,Y)