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 $\expect {X^3}$.

We find this using the moment generating function of $X$, $M_X$.

By Moment Generating Function of Gaussian Distribution, this is given by:


 * $\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$

From Moment in terms of Moment Generating Function:


 * $\expect {X^3} = \map {M_X'''} 0$

In Variance of Gaussian Distribution: Proof 2, it is shown that:

So:

Setting $t = 0$:

So: