Can $f(X)\times E+g(X)$ be a Gaussian if $E$ is a Gaussian for non-constant $f$ and $g$? It is well-known that $aE+b$ preserves the normality of $E$ when $a$ and $b$ are constants. However it is unclear whether this can happen when said constants become (possibly) dependent random variables.
So for simplicity, let $E\sim N(0,1)$ and $X$ is a continuous random variable independent of $E$, I want to know if there is some non-constant functions $f(x)$ and $g(x)$ such that $f(X)\times E+g(X)\sim N(0,1)$?

Your distribution is now that of two variables, $P(X,E)$, but for every value of $x$ the distribution of $E$, that is $P(E|X)$, is still Gaussian/Normal. It would be trickier, if $E$ and $X$ were correlated.
