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 for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \sigma^4 + 6 \sigma^2 \paren {\mu + \sigma^2 t}^2 + \paren {\mu + \sigma^2 t}^4} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}\) | simplifying |
$\blacksquare$