Skewness of Erlang Distribution

Theorem
Let $k$ be a strictly positive integer.

Let $\lambda$ be a strictly positive real number.

Let $X$ be a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$.

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


 * $\gamma_1 = \dfrac 2 {\sqrt k}$

Proof
From Skewness in terms of Non-Central Moments, we have:


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

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

By Expectation of Erlang Distribution we have:


 * $\mu = \dfrac k \lambda$

By Variance of Erlang Distribution we have:


 * $\sigma = \dfrac {\sqrt k} \lambda$

We also have:

So: