Computing $E[X|XY]$

Question

Let $X$ and $Y$ be independent continuous random variables with densities $f_{X}(x)$ and $f_{Y}(y)$. Also, let $Z = XY$. How does one compute $E[X|Z]$?

Answer 1

Lets assume $x$ and $y$ are continuous random variables and positive to simplify the problem. Then, the change of variables $(x,y)\rightarrow(x,xy)$ has Jacobian $\begin{pmatrix}1&0\\y&x\end{pmatrix}$ whose determinant absolute value is $x$ (positive). Therefore, $ f_{X,Z}(x,z)=f_X(x)f_Y(z/x)/x$. From here, $$ f_{X|Z}(x|z)=\frac{f_X(x)f_Y(z/x)}{x\int_0^\infty f_X(s)\frac{f_Y(z/s)}{s}ds}$$ from where we conclude $$E(X|Z)=\frac{\int_0^\infty f_X(x)f_Y(z/x)dx}{\int_0^\infty f_X(s)\frac{f_Y(z/s)}{s}ds} ~~.$$ I am not sure if this is what you were looking for.