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.
Best Answer
$Ee^{t\frac {Y-\mu} {\sigma}}=Ee^{t\frac Y {\sigma}} e^{-\frac {t\mu} {\sigma}}$. 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.