设随机变量(x,y)服从区域D={(x,y)|0

Question

设随机变量(x ,y)服从区域D={(x,y)|0

Answer 1

令Z=XY
显然D为边长为1的等腰直角三角形,其面积S=0.5
随机向量(X,Y)服从D上的二维均匀分布,
E(Z)=在D上(xy/S)即2xy的二重积分
cov(X,Y)=E(XY)-EX*EY
=E(Z)-EX*EY
=2 ∫ [下限0,上限1] x dx * ∫[下限0,上限x] ydy-∫ [下限0,上限1] x dx*∫ [下限0,上限1]ydy
=2 ∫ [下限0,上限1] 0.5x^3 dx-1/2*1/2
= (x^4)/4 [代入上下限1和0]-1/4
=