Suppose $X,Y$ are i.i.d. having density $f$ and $X,Y>-1$, I would like to calculate $$ E\left[ \frac{XY-1}{2+X+Y}\right]. $$
I was confusing about the following thinking: Because of i.i.d., am I allowed to write $$ E\left[ \frac{XY-1}{2+X+Y}\right] = ? =E\left[ \frac{X^2-1}{2+2X}\right] $$ I think not but not very clear about this. Any suggestion/comment is appreciated.
Just use the fact that $E(g(x,y))=\int_{-1}^{\infty}\int_{-1}^{\infty} \frac{xy-1}{2+x+y}f_{X,Y}(x,y) dy dx$ Now, use the fact that $f_{X,Y}(x,y)=f_X(x).f_Y(y)$