设X,Y,Z是三个随机变量,已知E(X)=E(Y)=1,E(Z)=-1;D(X)=D(Y)=D(Z)=2;P(X,Y)=0,P(Y,Z)=-0.5,P(Z,X)=0.5记W=X-Y+Z.试求E(W),D(W),E(W^2)

Question

设X,Y,Z是三个随机变量,已知E(X)=E(Y)=1,E(Z)=-1;D(X)=D(Y)=D(Z)=2;P(X,Y)=0,P(Y,Z)=-0.5,P(Z,X)=0.5
记W=X-Y+Z.试求E(W),D(W),E(W^2)

Answer 1

ρ(x,y)=cov(x,y)/(√D(x)√D(y))=[E(X,Y)-E(X)E(Y)]/2=0
cov(x,y)=0
同理
cov(x,z)=1
cov(y,z)=-1
E(W)=E(X)+E(Y)+E(Z)=1
D(W)=D(X+Y+Z)=D(X)+D(Y)+D(Z)+2cov(XY)+2cov(XZ)+2cov(YZ)=6+2-2=6
E(W²)=D(W)+E²(W)=7