① First, find the edge distribution density function of X and Y. According to the definition, fX(x)=∫(0, x)f(x, y)dy=4x? ，0 & ltx & lt 1; FX(x)=0, and x is other.

Similarly, fy (y) = ∫ (y, 1) f (x, y) dy = 4y (1-y? )，0 & lty & lt 1; FY(y)=0, and y is other.

② Seek the expected value. According to the definition, there is E(X)=∫(0, 1)xfX(x)dx=4/5. E(Y)=∫(0， 1)yfY(y)dy=(0， 1)4y？ ( 1-y？ )dy=8/ 15 .

E(XY)=∫(0， 1)∫(0，x)xyf(x，y)dxdy=∫(0， 1)dx∫(0，x)xyf(x，y)dy=8∫(0， 1)x？ dx∫(0，x)y？ dy=(8/3)∫(0, 1)(x^5)dx=4/9。

For reference.