Corollary 8.5.6. Gamma Properties via Moment Generating Function.
For the Gamma variable X,
\begin{equation*}
M(0) = 1
\end{equation*}
\begin{equation*}
M'(0) = r \mu
\end{equation*}
\begin{equation*}
M''(0) = r \mu^2 + \left ( r \mu \right )^2 = \sigma^2 + \mu^2
\end{equation*}