I have two independent random variables $X, Y$ with Poisson distribution say that $\mu$ and $\lambda$. I want to calculate the conditional expectation $\mathbb{E}\left(X | X^2 + Y^2\right)$.
It is known that $\mathbb{E}(X | X^2 + Y^2) = f(X^2 + Y^2)$ for a certain Borelian function $f$.
Can I tell something about $\mathbb{E}(X^2 | X^2 + Y^2)$ if i know $\mathbb{E}(X | X^2 + Y^2)$? Of course I know that $X^2 + Y^2 = \mathbb{E}(X^2 + Y^2 | X^2 + Y^2)$. Maybe this should helps.
Can I get some hints?