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

$\map { {M_X}^{\paren 4} } t = \paren {3 \sigma^4 + 6 \sigma^2 \paren {\mu + \sigma^2 t}^2 + \paren {\mu + \sigma^2 t}^4} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

## Proof

We have:

 $\ds \map { {M_X}^{\paren 4} } t$ $=$ $\ds \frac \d {\d t} \paren {3 \sigma^2 \paren {\mu + \sigma^2 t} + \paren {\mu + \sigma^2 t}^3} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$ Moment Generating Function of Gaussian Distribution: Third Moment $\ds$ $=$ $\ds \paren {3 \sigma^4 + 3 \sigma^2 \paren {\mu + \sigma^2 t}^2 } \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2} + \paren {3 \sigma^2 \paren {\mu + \sigma^2 t}^2 + \paren {\mu + \sigma^2 t}^4} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$ Chain Rule for Derivatives, Derivative of Power, Derivative of Exponential Function, Product Rule $\ds$ $=$ $\ds \paren {3 \sigma^4 + 6 \sigma^2 \paren {\mu + \sigma^2 t}^2 + \paren {\mu + \sigma^2 t}^4} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$ simplifying

$\blacksquare$