Question

已知X、Y分别服从正态分布N（0,9）和N（1,16）,且X与Y的相关系数ρXY=－1/2,设Z=X/3+Y/2,求

Answer 1

1）数学期望EZ=E(X/3+Y/2)=EX/3+EY/2=0+1/2=1/2

2）Y与Z的相关系数ρYZ

由ρXY=-1/2=[E(XY)-E(X)E(Y)]/[D(X)D(Y)]^0.5=[E(XY)-0*1]/3*4

所以E(XY)=-6

D(Z)=D(X/3+Y/2)=1/9*D(X)+1/4*D(Y)+2*1/3*1/2*ρXY*[D(X)D(Y)]^0.5

=1/9*3^2+1/4*4^2+2*1/3*1/2*(-1/2)*3*4

=3

ρYZ=[E(XZ)-E(X)E(Z)]/[D(X)D(Z)]^0.5

=[E(1/3X^2+XY/2)-0*1/2]/[3^2*3]^0.5

=[1/3*E(X)*E(X)+1/3*D(X)+1/2E(XY)]/(27)^0.5

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