A question about conditional expectation $E[X_1 | X_1 + X_2]$

Question

Let $X_1$ and $X_2$ be two iid random variables. I'm trying to find a formula for $E[X_1|X_1+ X_2]$. I was trying to show this $$E[X_1| X_1 + X_2 ]= \frac{E[X_1(X_1 + X_2)]}{E[X_1+ X_2]}$$ but I don't think it's true. Could you give me a tip to finda a counterexample of this?

Answer 1

It is not true.

$\qquad\mathsf E(X_1\mid X_1+X_2)$ is a random variable (a function of $X_1+X_2$), whereas the quotient is a constant.

For a quick example:

Let $X_1, X_2$ be Bernoulli($1/2$) random variables, so that:

$\qquad\mathsf P(X_1+X_2=z)~=~\begin{cases}1/4 &:& z=0\\1/2 &:& z=1\\1/4&:& z=2\\ 0&:& \text{otherwise}\end{cases}$

Then $\mathsf E(X_1\mid X_1+X_2)=\dfrac {(X_1+X_2)}{2}$ while $\dfrac{\mathsf E\big(X_1(X_1+X_2)\big)}{\mathsf E(X_1+X_2)}=\dfrac{3}{4}$ .

But any independent random variables will suffice.