Question

1.设随机变量X-N(1,16),Y-N(1,9),ρxy=0.5,令z=x/2+y/

Answer 1

1. 由ρxy=0.5得：Cov(X,Y)=ρxy*√(DX)*√(DY)=0.5*4*3=6

Cov(Y,Z)=Cov(Y,X/2+Y/3)=1/2*Cov(X,Y)+1/3*D(Y)=1/2*6+1/3*9=6

D(Z)=Cov(X/2+Y/3,X/2+Y/3)=Cov(X/2,X/2)+Cov(X/2,Y/3)+Cov(Y/3,X/2)+Cov(Y/3,Y/3)

=1/4*D(X)+1/6*Cov(X,Y)+1/6*Cov(X,Y)+1/9*D(Y)

=1/4*16+1/3*6+1/9*9=7

ρyz=Cov(Y,Z)/[√(DY)*√(DZ)]=6/[3*√7]=2/√7

2. E[(x-1)(x-2)]=E(X^2)-3E(X)+2=D(X)+E^2(X)-3E(X)+2

=λ+λ^2-3λ+2=λ^2-2λ+2=1

λ^2-2λ+1=0,λ=1