# Moment Generating Function of Gaussian Distribution/Examples/Third 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 third moment generating function of $X$ is given by:

$\map { {M_X}'''} t = \paren {3 \sigma^2 \paren {\mu + \sigma^2 t} + \paren {\mu + \sigma^2 t}^3} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

## Proof

We have:

 $\ds \map { {M_X}'''} t$ $=$ $\ds \frac \d {\d t} \paren {\paren {\sigma^2 + \paren {\mu + \sigma^2 t}^2} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2} }$ Moment Generating Function of Gaussian Distribution: Second Moment $\ds$ $=$ $\ds \paren {2 \sigma^2 \paren {\mu + \sigma^2 t} } \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2} + \paren {\sigma^2 \paren {\mu + \sigma^2 t} + \paren {\mu + \sigma^2 t}^3} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$ Chain Rule for Derivatives, Derivative of Power, Derivative of Exponential Function, Product Rule $\ds$ $=$ $\ds \paren {3 \sigma^2 \paren {\mu + \sigma^2 t} + \paren {\mu + \sigma^2 t}^3} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$ simplifying

$\blacksquare$