Conditional Expectation of A respect to A+B

Question

Let $A$ and $B$ be two independent random variables. Is there any bound for the following conditional expectation

$$\mathbb{E}[A|A+B].$$

Moreover, If we have $N$ independent random variables (non-identical) $A_i, i=1:N$, one can conclude that for large $N$: $$\mathbb{E}[A_1|\sum_{i=1}^{N}A_i]=\mathbb{E}[A_1].$$

Answer 1

$\mathbb E[A \mid A+B] + \mathbb E[B \mid A+B] = \mathbb E[A+B \mid A+B] = A+B$. If $B \ge b$, then $\mathbb E[B \mid A+B] \ge b$ so $\mathbb E[A \mid A+B] \le A+ B - b$. Similarly if $B \le c$, $\mathbb E[A \mid A+B] \ge A+B-c$.