Which functions of a normally distributed random variable are also normally distributed

Question

I know that if $X$ is normal then $Y$ = $f(X)$ = $aX + b$ is normal, and this is covered in other questions.

Are there any other cases?

Answer 1

Let $f$ be a continuous function such that $Y=f(X)$. Then, $$f_Y(y)=f_X(f^{-1}(y))\left|\frac{d}{dy}(f^{-1}(y))\right|\tag{1}$$ If $\ Y\sim\mathcal{N}(\mu,\sigma^2)$ where $X\sim\mathcal{N}(0,1)$, it follows that $$f_Y(y)=f_X\left(\frac{y-\mu}{\sigma}\right)\frac{1}{\sigma}\tag{2}$$ Comparing $(1)$ and $(2)$, it is seen that $f(y)=\sigma y+\mu$ is the only possible choice for $f$.