$X$ and $Y$ are independent and geometrical distributed with same parameter $p$. How to find $E(X|X+Y=k)$ for all $k =$ $2,3,4$....
I thought $$E(X|X+Y=k) = \sum_{x=1}^{k-1} xp(x,x+y=k)/p(x+y=k)$$ $n$ is infinity
now, I found $$p(x+y= k) = p^2 (1-p)^{k-2}$$ Am I on the correct path ? How do I find $p(x,x+y=k)$ ?