Corollary 8.5.4. Exponential Properties via Moment Generating Function.
For the Exponential variable X,
\begin{equation*}
M(0) = 1
\end{equation*}
\begin{equation*}
M'(0) = \mu
\end{equation*}
\begin{equation*}
M''(0) = \mu^2 + \mu^2 = \sigma^2 + \mu^2
\end{equation*}