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I have a model $Y = f(X) + \epsilon$

where $\epsilon$ is independent of $X$ and $\mathbb{E}[\epsilon]=0, \mathbb{E}\left[\epsilon^2\right]=\sigma^2$.

Show that $$ f(X)=\mathbb{E}[Y \mid X] $$

This is my attempt at the proof:

$$ \begin{aligned} & E[Y \mid X] \\ &= E[f(X)+\varepsilon \mid X] \\ &= E[f(X) \mid X]+E[\varepsilon \mid X] \\ &= E[f(X) \mid X]+E[\varepsilon] \\ &= E[f(X) \mid X]+0 \\ &= E_{Y \mid X}[f(X) \mid X] \quad\quad\quad \quad \quad\ \ (5) \\ &= \sum_{y \mid x} f(X) P(Y|X=y| x) \quad\quad(6) \\ &= f(X) \sum_{y \mid x} P(Y|X=y| x) \quad\quad(7)\\ &= f(X) \end{aligned} $$

I am a bit iffy about steps (5) to (6) to (7), please help. Thank you!

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    $\begingroup$ You were done before you reached step (5). In fact, at that point $Y$ no longer appears anywhere. You can't just stick $Y$ back in, either, because it involves $\epsilon,$ which also has vanished. A basic property of conditional expectations -- usually proven as soon as conditional expectation is defined -- is that $E[f(X)\mid X] = f(X).$ This is called "taking out what is known." $\endgroup$
    – whuber ♦
    Sep 30, 2022 at 22:18
  • $\begingroup$ Thank you! understood $\endgroup$
    – Cabbage Roll
    Oct 3, 2022 at 1:26

1 Answer 1

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$\mathbb{E}[Y|X] = \mathbb{E}[f(X)+\epsilon|X] = \mathbb{E}[f(X)|X]+\mathbb{E}[\epsilon|X] = \mathbb{E}[f(X)|X]+\mathbb{E}[\epsilon] = f(X)+0$

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    $\begingroup$ I think it is worth emphasizing that it is merely assumed that the conditional expectation of the true error (conditional on X) is zero, and that without this assumption, the theorem is not valid. $\endgroup$
    – Peter Leopold
    Oct 1, 2022 at 5:14
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    $\begingroup$ @PeterLeopold The OP asssumes that X and $\epsilon$ are independent. $\endgroup$
    – John Madden
    Oct 1, 2022 at 13:03
  • $\begingroup$ Thanks both, understood $\endgroup$
    – Cabbage Roll
    Oct 3, 2022 at 1:26

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