Moment Generating Function of Gaussian Distribution/Examples/Fourth Moment

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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

$\blacksquare$