Moment Generating Function of Gaussian Distribution

Theorem
Let $X \sim \Gaussian \mu {\sigma^2}$ 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:


 * $\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

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


 * $\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$

From the definition of a moment generating function:


 * $\ds \map {M_X} t = \expect { e^{t X} } = \int_{-\infty}^\infty e^{t x} \map {f_X} x \rd x$

So: