Conditional expectation of a random variable given the sum of two random variables

Question

Is there any possibility to simplify the expression $\mathbb{E}[Y|(X_{1} + X_{2})]$? Assume that $\mathbb{E}[Y]$, $\mathbb{E}[X_{1}]$, $\mathbb{E}[X_{2}]$, $\mathbb{E}[(X_{1} + X_{2})]$ all exist. Is there a difference if $X_{1}$ and $X_{2}$ are not independent?

Can I simply write $\mathbb{E}[Y|(X_{1} + X_{2})] = \mathbb{E}[Y|X_{1}] + \mathbb{E}[Y|X_{2}]$?

Answer 1

Can I simply write $E[Y|(X1+X2)]=E[Y|X1]+E[Y|X2]?$

Not generally.

Counterexample

Let $X_1, X_2$ be the result of two independent dice throws (six-sided, enumerated, unbiased).   Further, let $Y:=X_1+X_2$.

Then

• $\mathsf E(Y\mid X_1+X_2)= X_1+X_2$
• $\mathsf E(Y\mid X_1) = X_1+\mathsf E(X_2)$
• $\mathsf E(Y\mid X_2) = \mathsf E(X_1)+X_2$
• $\mathsf E(X_1)+\mathsf E(X_2)=7$

Therefore: $\mathsf E(Y\mid X_1+X_2) =\mathsf E(Y\mid X_1)+\mathsf E(Y\mid X_2)-7$