Corollary 9.5.4. Chi-Square Properties via Moment Generating Function.
For the Chi-Square variable X,
\begin{equation*}
M(0) = 1
\end{equation*}
\begin{equation*}
M'(0) = r = \mu
\end{equation*}
\begin{equation*}
M''(0) = 2r + r^2 = \sigma^2 + \mu^2
\end{equation*}