# Finding the Moment Generating Function of Standard Normal Random Variable from Normal Random Variable

Given $$Y\sim N(\mu, \sigma^2)$$. I'm trying to find the moment generating function of $$Z=\frac{Y-\mu}{\sigma}$$ using the MGF transform method.

Here's what I've tried:
$$M_Y(t)=e^{\mu t+\frac{\sigma^2t^2}{2}}$$ $$M_Z(t) = E(e^{Zt}) = E(e^{t\frac{Y-\mu}{\sigma}})$$ and I'm stuck completely. I'm thinking that Z follows a standard normal distribution, hence the resulting MGF of it will be: $$M_Z(t) = e^{\frac{1}{2}t^2}$$ However, I do not know how to get there from where I'm stuck at. Can anyone help me with this? Thank you.

$$Ee^{t\frac {Y-\mu} {\sigma}}=Ee^{t\frac Y {\sigma}} e^{-\frac {t\mu} {\sigma}}$$. Note that $$e^{-\frac {t\mu} {\sigma}}$$ is a constant and it can be pulled out of the expectation. Now $$Ee^{t\frac Y {\sigma}}$$ is nothing but $$Ee^{s Y }$$ where $$s=\frac t {\sigma}$$. Use the formula you have for $$M_Y(s)$$ to finish.