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I have the random variables $X$ and $Y$ related through the following: \begin{align} X &= N_X \\ Y &= 4X + N_Y \end{align} where $N_X, N_Y \overset{\text{iid}}{\sim} \mathcal N(0,1)$. Is there a faster way to derive the conditional density $f_{X|Y=y}(x)$ than computing the joint distribution and dividing it by the marginal distribution of $Y$? For reference, for $y=2$, we should obtain $f_{X|Y=2}(x) = \mathcal N(8/17, 1/17)$.

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    $\begingroup$ This is classical linear regression, so if you are familiar with the theory and formulas you can immediately write down the answer--but if not, then developing that theory would not be any faster! $\endgroup$ – whuber Jun 25 at 18:49
  • $\begingroup$ which expression of linear regression do you have in mind? $\endgroup$ – Gepeto97 Jun 25 at 20:17
  • $\begingroup$ Ordinary least squares. $\endgroup$ – whuber Jun 25 at 20:35

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