Theorem 9.5.1. Moment Generating Function for Normal. Presuming \(t \gt 0\) and \begin{equation*} M(t) = e^{t \mu+\frac{1}{2}t^2\sigma^2} \end{equation*}
Corollary 9.5.2. Normal Properties via Moment Generating Function. For the Normal variable X, \begin{equation*} M(0) = 1 \end{equation*} \begin{equation*} M'(0) = \mu \end{equation*} \begin{equation*} M''(0) = \sigma^2 + \mu^2 \end{equation*}