Skewness of Pareto Distribution

Theorem
Let $X$ be a continuous random variable with the Pareto distribution with $a, b \in \R_{> 0}$.

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


 * $\gamma_1 = \begin {cases} \paren {\sqrt {\dfrac {\paren {a - 2} } a } } \paren { \dfrac {2 \paren {a + 1} } {a - 3} } & 3 < a \\ \text {does not exist} & 3 \ge a \end {cases}$

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 Pareto Distribution we have:


 * $\mu = \dfrac {a b } {\paren {a - 1} }$

By Variance of Pareto Distribution we have:


 * $\sigma = \dfrac {\sqrt a b } {\sqrt {\paren {a - 2} } \paren {a - 1} }$

From Raw Moment of Pareto Distribution, we have:


 * $\ds \expect {X^3} = \begin {cases} \dfrac {a b^3} {a - 3} & 3 < a \\ \text {does not exist} & 3 \ge a \end {cases}$

So: