Conditional expectation $E(X_1 \mid \overline{X}_n)$ if $X_1,\dots,X_n$ are i.i.d. Am I correct?

Question

Conditional expectation $E(X_1 \mid \overline{X}_n)$ if $X_1,\dots,X_n$ are i.i.d.

Since $X_1,\dots,X_n$ are i.i.d, then $E(X_1 \mid \overline{X}_n) = E(X_1)=\overline{X}_n$

Am I correct in thinking this? Thanks!

(Just to overclearify $\overline{X}_n$ is the sample mean) Question is from Van Der Vaart: Asymptotic Statistics.

Answer 1

We have $$E(X_i |\bar{X}_n)= E(X_j |\bar{X}_n)$$

Summing them up $$\sum_{i=1}^n E(X_i|\bar{X}_n)=E(\sum_{i=1}^n X_i|\bar{X}_n)=E(n\bar{X}_n|\bar{X}_n)=n\bar{X}_n$$

$$nE(X_1|\bar{X}_n)=n\bar{X}_n$$ Hence, $$E(X_1|\bar{X}_n)=\bar{X}_n$$