Skewness of Bernoulli Distribution

Theorem
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$.

Then the skewness $\gamma_1$ of $X$ is given by:


 * $\gamma_1 = \dfrac {1 - 2 p} {\sqrt {p q} }$

where $q = 1 - p$.

Proof
From Skewness in terms of Non-Central Moments:


 * $\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$

where $\mu$ is the mean of $X$, and $\sigma$ the standard deviation.

We have, by Expectation of Bernoulli Distribution:


 * $\mu = p$

By Variance of Bernoulli Distribution, we also have:


 * $\var X = \sigma^2 = p \paren {1 - p}$

so:


 * $\sigma = \sqrt {p \paren {1 - p} }$

To now 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 Bernoulli Distribution, this is given by:


 * $\map {M_X} t = q + p e^t$

By Moment in terms of Moment Generating Function:


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

In Variance of Bernoulli Distribution: Proof 5, it is shown that:


 * $\map {M_X''} t = p e^t$

So, by Derivative of Exponential Function:


 * $\map {M_X'''} t = p e^t$

Setting $t = 0$:

So: