Skewness of Erlang Distribution
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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:
\(\ds \expect {X^3}\) | \(=\) | \(\ds \frac 1 {\lambda^3} \prod_{m \mathop = 0}^2 \paren {k + m}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {k \paren {k + 1} \paren {k + 2} } {\lambda^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {k^3 + 3 k^2 + 2 k} {\lambda^3}\) |
So:
\(\ds \gamma_1\) | \(=\) | \(\ds \frac {\lambda^3} {k^{3 / 2} } \paren {\frac {k^3 + 3 k^2 + 2 k} {\lambda^3} - \frac {3 k^2} {\lambda^3} - \frac {k^3} {\lambda^3} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\lambda^3} {k^{3 / 2} } \cdot \frac {2 k} {\lambda^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt k}\) |
$\blacksquare$