Question

三个自由变量Z,X,Y满足z=x+y,协方差 c(z,y)可以等于0吗?

Answer 1

c(z,y)=E{[Z-E(Z)][Y-E(Y)]}/[D(Z)D(Y)]^0.5 (1)

其中：E(Z)=E(X)+E(Y) (2)

D(Z)=E{[Z-E(Z)]^2}=E{[X-E(X)+Y-E(Y)]^2}=D(X)+D(Y)+E{2[X-E(X)][Y-E(Y)]}

D(Z)=σ²(Z)=D(X)+D(Y)+2σ²(X,Y) (3)

记：σ(z)σ(y)=[D(Z)D(Y)]^0.5 (4)

c(z,y)=E{[X+Y-E(X)-E(Y)][Y-E(Y)]}/[σ(z)σ(y)]

=E{[X-E(X)][Y-E(Y)]+[Y-E(Y)]^2}/[σ(z)σ(y)]

={σ(x)σ(y)c(x,y)+σ²(y)}/[σ(z)σ(y)]

={σ(x) c(x,y)+σ(y)}/σ(z) (5)

可见：一般c(z,y) 不为0,除非X,Y独立,且D(Y)=0；或者X,Y的方差都为0.