Skewness of Log Normal Distribution

Theorem
Let $X$ be a continuous random variable with the Log Normal distribution with $\mu \in \R, \sigma \in \R_{> 0}$.

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


 * $\gamma_1 = \paren {\exp {\paren {\sigma^2} } + 2} \sqrt {\paren {\exp {\paren {\sigma^2 } } - 1} } $

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 Log Normal Distribution we have:


 * $\mu = \exp {\paren {\mu + \dfrac {\sigma^2 } 2 } }$

By Variance of Log Normal Distribution we have:


 * $\sigma = \exp {\paren {\mu + \dfrac {\sigma^2} 2 } } \sqrt {\paren {\exp {\paren {\sigma^2  } } - 1} }$

From Raw Moment of Log Normal Distribution, we have:


 * $\expect {X^3} = \exp {\paren {3\mu + \dfrac {3^2 \sigma^2 } 2 } }$

So: