1.构造实变量t的二次函数
f(t)=E(Xt-Y)^2
=E(X^2*t^2-2*X*Y*t+Y^2)
=t^2*E(X^2)-2*E(X*Y)*t+E(Y^2)
∵对任意的t，f(t)>=0
∴4*(E(X*Y))^2-4*E(X^2)*E(Y^2)<=0
即(E(X*Y))^2<=E(X^2)*E(Y^2) (1)
2.令X1=X-EX,Y1=Y-EY
则E(X1)=0,E(Y1)=0
[cov(X,Y)]^2=[E(X-EX)(Y-EY)]^2
=[E(X1*Y1)]^2
<=E(X1^2)*E(Y1^2) 由(1)式
=[E(X1^2)-(E(X1)^2)]*[E(Y1^2)-(E(Y1)^2)]
=D(X1)*D(Y1)
=D(X-EX)*D(Y-EY)
=D(X)*D(Y)