Conditional Expectation of a discrete random variable given a sum involving the random variable.

Question

Fix $ n{\geq}2$. Let $ X_{1},X_{2},...,X_{n}$ be n i.i.d discrete random variables with finite mean. Set $ S_{n}:= \sum\limits_{i=1}^n X_{i} $. Show that $ E[X_{1}|S_{n}] = \frac{S_{n}}{n}$.

Answer 1

Hint: What about $E[X_{2}|S_{n}]$? or $E[X_{3}|S_{n}]$? Then, we can also utilize the linearity of expectation for $E[X_{1}+\dots+X_{n}|S_{n}] = \dots?$.

Answer 2

Since $ X_{1},....,X_{n} $ are i.i.d, we have that $ E[X_{1}|S_{n}]=E[X_{2}|S_{n}]=...=E[X_{n}|S_{n}] $. Of course, $ E[X_{1}+...+X_{n}| S_{n}] = S_{n}$. By the linearity property of condition expectation $$E[X_{1}+...+X_{n}|S_{n}]=\sum\limits_{i=1}^nE[X_{i}|S_{n}] $$ Therefore we must have that $$ E[X_{i}|S_{n}]=\frac{S_{n}}{n} $$