$\mathbb E[X_i\mid X_1,...,X_n]=\mathbb E[X_i]$

Question

Do you agree that if $(X_i)$ is a sequence of i.d.d. random variable, then for all $i$

$$\mathbb E[X_i\mid X_1,...,X_n]=\mathbb E[X_i]\ \ \ ?$$

Answer 1

I don't agree. Given the outcomes of all $X_i$ you know exactly what the conditional expectation is, namely, $X_i$ for $i \leq n$. $$E[X_i | X_1, ..., X_n] = X_i$$ That's true regardless of if the sequence are iid or not.

Since the $X_i$ are independent, this does hold true for $i > n$ $$E[X_i | X_1, ..., X_n] = E[X_i]$$