Moment Generating Function Normal Distribution

Question

I'm very stuck on this question, I'm not sure where I'm supposed to start. Any help would really help

Answer 1

• Look up the MGF for the $N(\mu, \sigma^2)$ and $N(0,1)$ distributions.
• The MGF of $Z$ is $M_Z(t) = E[e^{tZ}]$ by definition. Plugging in the definition of $Z$ yields $E[e^{t(X-\mu)/\sigma}] = E[e^{(t/\sigma) X}] e^{-\mu t/\sigma} = M_X(t/\sigma) e^{-\mu/t}$. Plug in the formula for $M_X$ and check that the end result is the MGF of the $N(0,1)$ distribution.