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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?

Nourless
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    From https://math.stackexchange.com/q/78546/321264, you can see that $E\left[X_k\mid \sum_{k=1}^n X_k\right]=\frac1n\sum_{k=1}^n X_k$ for every $k=1,\ldots,n$. – StubbornAtom Jun 06 '21 at 15:42

1 Answers1

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Hint:

For i.i.d samples:

$\mathbb{E}(X_1|\sum\limits_{k=1}^n X_k)=\mathbb{E}(X_2|\sum\limits_{k=1}^n X_k)=...=\mathbb{E}(X_n|\sum\limits_{k=1}^n X_k)$

$\mathbb{E}(\sum\limits_{k=1}^n X_k|\sum\limits_{k=1}^n X_k)=\sum\limits_{k=1}^n X_k$

Masoud
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