Let $X_1, \dots, X_n$ i.i.d random variables. Put $S_n:= \sum_{k=1}^n X_k$.
Show that $\mathbb{E}[X_1 I_{\{S_n \in A\}}]= \mathbb{E}[X_j I_{\{S_n \in A\}}]$ for $1 \leq j \leq n$, where $A$ is an arbitrary Borel set.
A possible idea of mine was to prove that the random vectors $(X_1, S_n)$ and $(X_j, S_n)$ have the same joint distribution. The statement intuitively looks obvious because the sum $S_n$ is symmetric in all variables $X_1, \dots, X_n$ so it shouldn't matter which variable we write on the front.
How can I formally show this? (Fubini and all these theorems like are allowed). These things are always hard to show yet seem so intuitive.
Best Answer
The i.i.d. random variables have this property: Let $\pi$ be any permutation of $\{1,\dots,n\}$. Then the two vectors $$ (X_1,X_2,\dots,X_n)\qquad \text{and}\qquad (X_{\pi(1)},X_{\pi(2)},\dots,X_{\pi(n)}) $$ have the same joint distribution. This is called exchangeable. See that wiki site for more information.