Theorem 9.5.3. Moment Generating Function for Chi-Square. Presuming \(t \gt 0\) and \begin{equation*} M(t) = \left ( \frac{1}{1-2t} \right )^{r/2} \end{equation*}
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*}