Corollary 7.5.3. Geometric Properties via Moment Generating Function.
For the Geometric variable X,
\begin{equation*}
M(0) = 1
\end{equation*}
\begin{equation*}
M'(0) = \frac{1}{p} = \mu
\end{equation*}
\begin{equation*}
M''(0) = \frac{1-p}{p^2} + \frac{1}{p^2} = \sigma^2 + \mu^2
\end{equation*}