During lecture my professor worked the example of $E(X\mid X+Y)$, where $X$ and $Y$ are independent Poisson random variables. So in an effort to prepare for the test![ this is not a graded or assigned problem just for doing it on my own] I decided to try and compute $E(X\mid X-Y)$, I also tried and failed in computing $E(X\mid XY)$, using the same technique, but I am running into trouble.
If you could give any insight on how I should proceed, or even just a better way to approach this type of problem. I attached the work I did on $E(X\mid X-Y)$, but if you want to work $E(X\mid XY)$ that is fine with me. I appreciate any and all help.
Thanks in advance.