Skewness of Gaussian Distribution/Proof 2

Proof
From the definition of skewness, we have:


 * $\gamma_1 = \expect {\paren {\dfrac {X - \mu} \sigma}^3}$

where:
 * $\mu$ is the expectation of $X$.
 * $\sigma$ is the standard deviation of $X$.

By Expectation of Gaussian Distribution, we have:


 * $\mu = \mu$

By Variance of Gaussian Distribution, we have:


 * $\sigma = \sigma$

So:

To calculate $\gamma_1$, we must calculate $\expect {X^3}$.

From Moment in terms of Moment Generating Function:
 * $\expect {X^n} = \map { {M_X}^{\paren n} } 0$

where $M_X$ is the moment generating function of $X$.

From Moment Generating Function of Gaussian Distribution: Third Moment:


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

Setting $t = 0$:

So: