Chapter 20.1, Number 11
Show that if X is a random variable with
\[ u=mean\\ \sigma^{2}=variance \]
and if
\[ X^{*}=\frac{X-u}{\sigma} \]
is the standardized version of X, then
\[ gx*(t)=e^{-\frac{ut}{\sigma}}(\frac{t}{\sigma}) \]
This problem sounds harder than it is if you consider what we are actually given.
We know
\[ X^{*}=\frac{X-u}{\sigma} \] I would encourage you to check this video because it is a really good breakdown on moment generating functions https://www.youtube.com/watch?v=cbmfYoepHPk
The moment generating function for g(t) is computed as follows:
\[ g_x(t)=E(e^{tX^{*}})\\ =E(e^{t\frac{X-u}{\sigma}})\\ =E(e^{\frac{tX}{\sigma}}e^{\frac{-tu}{\sigma}})\\ =e^{-t\frac{u}{\sigma}}E(e^{t\frac{X}{\sigma}})\\ =e^{-t\frac{u}{\sigma}}g_x(t) \]