Given i.i.d. $\{X_k\}_{k=1}^n$ I wonder how I can compute $$ \mathbb{E}(X_1 + 2X_2|\sum\limits_{k=1}^n X_k). $$ I realize that I should expand it as $$ \mathbb{E}(X_1|\sum\limits_{k=1}^n X_k) + 2\mathbb{E}(X_2|\sum\limits_{k=1}^n X_k). $$ But what to do next? How can I extract knowledge from the sum?
2026-04-24 06:13:46.1777011226
How to compute conditional expectation given sum of variables?
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Hint:
For i.i.d samples:
$\mathbb{E}(X_1|\sum\limits_{k=1}^n X_k)=\mathbb{E}(X_2|\sum\limits_{k=1}^n X_k)=...=\mathbb{E}(X_n|\sum\limits_{k=1}^n X_k)$
$\mathbb{E}(\sum\limits_{k=1}^n X_k|\sum\limits_{k=1}^n X_k)=\sum\limits_{k=1}^n X_k$