设二维随机变量（X,Y）的概率密度为：f(x,y)=12y^2,0<=y<=x<=1;f(x,y)=0,其他，求E(X),E(Y),E(X^2+Y^2)。

Question

希望给出详细解答，谢谢各位……

Answer 1

EX=∫∫[0<=y<=x<=1] xf(x,y)dxdy=∫[0->1]∫[0->x] 12xy²dydx=4/5
EY=∫∫[0<=y<=x<=1] yf(x,y)dxdy=∫[0->1]∫[0->x] 12y³dydx=3/5
E(X²+Y²)=∫∫[0<=y<=x<=1] (x²+y²)f(x,y)dxdy=∫[0->1]∫[0->x] 12x²y²+12y^4dydx=16/15

Answer 2

EX=∫∫[0<=y<=x<=1] xf(x,y)dxdy=∫[0->1]∫[0->x] 12xy²dydx=4/5
EY=∫∫[0<=y<=x<=1] yf(x,y)dxdy=∫[0->1]∫[0->x] 12y³dydx=3/5
E(X²+Y²)=∫∫[0<=y<=x<=1] (x²+y²)f(x,y)dxdy=∫[0->1]∫[0->x] 12x²y²+12y^4dydx=16/15