Conditional Expectation of Poisson Distribution

Question

So I am not sure how to go about this,

Say that $X_j\sim$Pois$(\theta)$, and are iid. Find the following: $$ E[X_1+2X_2+3X_3|\sum_{j=1}^nX_j] $$

I am aware that I am suppose to somehow make use of the fact that $\sum_{j=1}^nX_j\sim$Pois$(n\theta)$, but I am not sure how to apply it here.

Answer 1

Lemma: If $X_1,X_2,...,X_n$ are iid such that $E[X_i]<\infty$, and $S_n=\sum_{j=1}^nX_j$ then $E[X_i|S_n]=\frac{1}{n}S_n$ for all $i\in\{1,...,n\}$

Proof: $S_n=E[S_n|S_n]=E[\sum_{j=1}^nX_j|S_n]=\sum_{j=1}^nE[X_j|S_n]=nE[X_i|S_n]$

Now, for your problem,

$\begin{eqnarray}E[X_1+2X_2+3X_3|S_n]&=&E[X1|S_n]+2E[X_2|S_n]+3E[X_3|S_n]\\&=&\frac{1}{n}S_n+\frac{2}{n}S_n+\frac{3}{n}S_n\\&=&\frac{6}{n}S_n\end{eqnarray}$