# Math Help - Conditional expectation

1. ## Conditional expectation

How can I prove that if
$(X_{n})_{n \geq 1}$ are i.i.d. and $S_{n}=X_{1}+X_{2}+...+X_{n}$
then
$\mathbb{E}(X_{1}|S_{n},S_{n+1},...)=\frac{S_{n}}{n }, \forall n=1,2,...$?

2. Hello,
Originally Posted by yavanna
How can I prove that if
$(X_{n})_{n \geq 1}$ are i.i.d. and $S_{n}=X_{1}+X_{2}+...+X_{n}$
then
$\mathbb{E}(X_{1}|S_{n},S_{n+1},...)=\frac{S_{n}}{n }, \forall n=1,2,...$?
First prove that $\forall k=1\dots n ~,~ \text{ the } (X_k,S_n)$ follow the same distribution
(not very difficult since the rv's are iid)

Then we have, for any bounded (measurable) function f, and for $j\neq k$, $E[E[X_k|S_n]f(S_n)]=E[X_kf(S_n)]=E[X_jf(S_n)]=E[E[X_j|S_n]f(S_n)]$

So $E[X_k|S_n]=E[X_j|S_n]$ almost surely.

Thus $E[X_k|S_n]=\frac 1n \cdot \sum_{j=1}^n E[X_j|S_n]=\frac 1n \cdot E[S_n|S_n]=\frac{S_n}{n}$