Assume $X,Y$ to be random variables defined on $(\Omega,\Sigma,P)$, $Y = aX + b$ and that $X$ is Gaussian. I wanted to derive the conditional p.d.f. defined by:
$f_{Y|X}(y|x) := \frac{f_{X,Y}(x,y)}{f_X(x)}$
It is easy to show that $E[Y|X] = Y$ a.s. and also if $g(x) = \int_\mathbb{R} yf_{Y|X}(y|x)dy$ then $(g \circ X)(\omega) = g(X(\omega))$ is also a version of the conditional expectation (This is shown in the textbook titled "Probability With Martignales" by David Williams). Therefore,
$g(X(\omega)) = aX(\omega) + b\text{ a.s.}$
Is it possible to derive $f_{Y|X}(y|x)$ from the above relation? If not, what is the best way to derive this?
Any help is much appreciated. Thanks