设随机变量（X,Y)的联合概率密度为f(x,y)=8xy,0<x<1,0<y<x;0,其他，

情况1: x<0,y<0
都未开始积分
F(x,y)=0

情况2: 0<x<1,0<y<x
都未积完
F(x,y)
=∫(0~y)∫(y~x) 8xy dxdy
=∫(0~y) 4(x²-y²)y dy
=4(x²y²/2-y^4/4)
=2(x²-y²)y²

情况3: 0<x<1,y>x
x未积完，y积完
F(x,y)
=Fx(x)
=∫(0~x)∫(0~x) 8xy dydx
=∫(0~x) 4xx² dx
=x^4

情况3: x>1, 0<y<x
x积完，y未积完
F(x,y)
=Fy(y)
=∫(0~y)∫(y~1) 8xy dxdy
=∫(0~y) 4(1-y²)y dy
= 2y²-y^4

情况4: x>1,y>1
全部积完
F(x,y)=1

fy(y)=∫(y~1) 8xy dx
=4(1-y²)y

fx|y(x|y)=f(x,y)/fy(y)
=8xy/4(1-y²)y
=2x/(1-y²)
0<x<1,0<y<x
=0 其他

E(X|Y)
=∫(y~1) x*fx|y(x|y) dx
=∫(y~1) 2x²/(1-y²) dx
=∫(y~1) 2x³/(3(1-y²)) dx
=(2/3)(1+y+y²)/(1+y)

P(0<=y<2x<=1)

0<=y<2x,但是y<x

=∫(0~1/2)∫(0~x) 8xy dydx
= ∫(0~1/2) 4x(x²) dx
=x^4 (0~1/2)
=1/16