设二维连续随机向量(X，Y)的概率密度
f(x,y)={1/8(x+y)0≤x≤2,0≤y≤2；
{0其他．
求E(X)，E(Y)，Cov(X，Y)，ρXY
【正确答案】：E(X)=∫+∞-∞∫+∞-∞xf(x，y)dxdy ， =∫20x[∫201/2(x+y)dy]dx=6/7， E(Y)=∫20∫20[y•1/2(x+y)dy]dx=6/7． Cov(X，Y)=E[(x-7/6)(Y-7/6)] =∫20∫20(x-7/6)(y-7/6)1/8(x+y)dxdy =∫20∫20[xy-(7/6)x-(7/6)y+(7/6)2]1/8(x+y)dy}dx =-(1/36) D(X)=E[X-E(X)]2 =∫+∞-∞∫+∞-∞[x-E(X)]2f(x，y)dxdy =∫20∫20(x-7/6)21/8(x+y)dxdy=11/36． D(Y)=11/36． 所以ρXY=Cov(X,Y)/√D(x)=D(Y)=[-(1/36)/11/36]=-(1/ 11)．
