Expectation of function of random variables

Question

I have two independent random variables, X and Y. U and V are defined as U=X+Y and V=XY. I need to find Covariance. I know, E(U)= E(X)+E(Y) and E(V)=E(XY)=E(X)E(Y). But how to write E(UV) in terms of E(X) and E(Y). Any help is appreciated.

Answer 1

Note that $$ VU=XY(X+Y)=X^2Y+XY^2 $$ aso $$ EUV=EX^2Y+EXY^2 $$ by linearity and $$ EX^2Y=EX^2EY; \quad EXY^2=EXEY^2 $$ since $X, Y$ are independent and functions of independent random variables are independent.