A little confusion about conditional expectation

Question

If $(X_i)$ are iid. Let $S_n=\sum^n_{i=1}X_i$, then how do we compute $E(X_i\mid S_n)$. Is it independent of $i$?

I know it is a random variable. I guess that is independent of $i$, but I don't know how to show.

Answer 1

The fact that $E(X_i\mid S_n)=E(X_j\mid S_n)$ is obvious by symmetry.

For the follow-up question about $E(X_i\mid S_n)$, note that by the linearity of (conditional) expectation we have $$E((X_1+\cdots+X_n)\mid S_n)=E(X_1\mid S_n)+\cdots +E(X_n\mid S_n).$$ But $E((X_1+\cdots +X_n)\mid S_n)=E(S_n\mid S_n)=S_n$. It follows that $E(X_i\mid S_n)=\frac{S_n}{n}$.