Moment Generating Function of Gaussian Distribution/Examples/Second Moment

Examples of Use of Moment Generating Function of Gaussian Distribution
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution.

The second moment generating function of $X$ is given by:


 * $\map { {M_X}''} t = \paren {\sigma^2 + \paren {\mu + \sigma^2 t}^2 } \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

Proof
We have: