It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the random variable. So I hope you can help me remember / figure out how to solve it.
Let $x$ be a normally distributed random variable. Now, consider the random variable $y = g(x)$, where $g(x) = 1$ if $x \leq a$, $g(x) = 2$ if $a< x \leq b$ and $g(x) = 3$ if $x > b$, and $a,b$ are some constants in $R$.
My goal is to see if the final distribution belongs to the exponential family, and knowing the distribution would be a necessary first step. If the function was continuous, I would know what to do, but I am somewhat stuck since it is not. Any help including links to theorems is appreciated!