Another exam review question.
Suppose I know $Z \sim \text{N}(0,1)$, $Y=|Z|$. And I want to find the moment generating function $M_{Y}(t)$. In our notes it's just given to us as $2 N(t) e^{\frac{t^{2}}{2}}$ where $N(t)$ is the c.d.f. of a $N(0,1)$ random variable, but I felt like for practicing mgfs it could be good to derive it. That being said, I don't even know where to start. I feel like starting with the normal p.d.f., but from there I don't know where to simplify, if that's the right approach. Maybe MacLaurin series?