EXERCISE 3.68 in Hsu's book

First calculate the marginal PDF's of X and Y according to formulas 3.30-3.31

Clear[fX,fY,fXY];fXY[x_,y_]:=Exp[-(2/3)*(x^2-x*y+y^2)]/(Sqrt[3]*Pi);fX[x_]:=Integrate[fXY[x,y],{y,-Infinity,Infinity}];fY[y_]:=Integrate[fXY[x,y],{x,-Infinity,Infinity}];

Now calculate the asked for quantities in the exercise according to their definitions

Clear[EX,EY,VarX,VarY,EXY,rhoXY];EXY=Integrate[y*Integrate[x*fXY[x,y],{x,-Infinity,Infinity}],{y,-Infinity,Infinity}];{EX,EY,VarX,VarY,rhoXY}={Integrate[x*fX[x],{x,-Infinity,Infinity}],Integrate[y*fY[y],{y,-Infinity,Infinity}],Integrate[x^2*fX[x],{x,-Infinity,Infinity}]-EX^2,Integrate[y^2*fY[y],{y,-Infinity,Infinity}]-EY^2,(EXY-EX*EY)/(Sqrt[VarX*VarY])}