Compute $\mathbb{E}(X_1\mid S_n=a)$

Question

Suppose $X_1, X_2, \dots, $ be i.i.d. with finite mean and assume $S_n=\sum_{i=1}^n X_i$. Compute $$\mathbb{E}(X_1\mid S_n=a).$$

I know that $$\mathbb{E}(X_1\mid S_n)=\frac{S_n}{n}.$$ Does it work?

Answer 1

Answer is correct. Let me add in more details.

$$\sum_{i=1}^n \mathbb{E}(X_i|S_n) = \mathbb{E}(\sum_{i=1}^nX_i|S_n)= \mathbb{E}(S_n|S_n)=S_n$$

by IID, we have $\mathbb{E}(X_i|S_n)=\mathbb{E}(X_1|S_n)$,

Hence $$n \mathbb{E}(X_1|S_n)=S_n$$

$$ \mathbb{E}(X_1|S_n)=\frac{S_n}n$$