# Conditional Gaussian from Joint Distribution

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 $$f_{X|Y=2}(x) = \mathcal N(8/17, 1/17)$$.

• 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! – whuber Jun 25 at 18:49
• which expression of linear regression do you have in mind? – Gepeto97 Jun 25 at 20:17
• Ordinary least squares. – whuber Jun 25 at 20:35