Moment Generating Function of Gaussian Distribution

Theorem
Let $X \sim N \left({\mu, \sigma^2}\right)$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution.

Then the moment generating function $M_X$ of $X$ is given by:


 * $M_X \left({t}\right) = \exp \left({\mu t + \dfrac 1 2 \sigma^2 t^2}\right)$

Proof
From the definition of the Gaussian distribution, $X$ has probability density function:


 * $f_X \left({x}\right) = \dfrac 1 {\sigma \sqrt{2 \pi} } \, \exp \left({-\dfrac { \left({x - \mu}\right)^2} {2 \sigma^2} }\right)$

From the definition of a moment generating function:


 * $\displaystyle M_X \left({t}\right) = \mathbb E \left[{ e^{t X} }\right] = \int_{-\infty}^\infty e^{tx} f_X \left({x}\right) \rd x$

So: