We can show that the regression for two variables can be written as
$$ E(\mathbf{Y} \mid \mathbf{X}=\mathbf{x}) = \alpha + \beta X $$
So for example if I consider let the gaussian vector Z=(X,Y) It means we have
So
Is my reasoning correct ?
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$\begingroup$ If your question is about given $(X, Y)$ is jointly Gaussian, what $\alpha$ and $\beta$ are in terms of $\mu$ and $\Sigma$, then your reasoning is correct. However, $E[Y|X = x]$ most of the time is not a linear function of $x$, see this thread for more detailed discussions. $\endgroup$– ZhanxiongCommented Feb 20, 2023 at 21:33
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Your reasoning is correct.
You can also rearrange the expression a bit to obtain the linear update formula $$ \mathbb{E}(Y| X=x) = \mathbb{E}(Y) + \mathrm{Cov}(Y,X) \mathrm{Var}(X)^{-1}[x-\mathbb{E}(X)] $$
Before we observe $X$, our best estimate of $Y$ is $\mathbb{E}(Y)$. Once we observe $X = x$, we can update our estimate with the formula.
