设二维连续随机变量(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)，pXY
E(X)=∫+∞-∞∫+∞-∞xf(x,y)dxdy =∫20x[∫201/2(x+y)dy]dx=6/7； E(Y)=∫20[∫20y•1/8(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pXY=Cov(X,Y)/√D(X)√D(Y)=-(1/36)/(11/36)=-(1/11).
