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:

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

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:

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

By Variance of Log Normal Distribution:

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

From Raw Moment of Log Normal Distribution:

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

So:

\(\ds \gamma_1\)

\(=\)

\(\ds \frac {\map \exp {3 \mu + \dfrac {9 \sigma^2} 2} - 3 \map \exp {\mu + \dfrac {\sigma^2} 2} \paren {\map \exp {2 \mu + \sigma^2} \paren {\map \exp {\sigma^2} - 1} } - \paren {\map \exp {\mu + \dfrac {\sigma^2} 2} }^3} {\paren {\map \exp {\mu + \dfrac {\sigma^2} 2} \sqrt {\paren {\map \exp {\sigma^2} - 1} } }^3}\)

\(\ds \)

\(=\)

\(\ds \frac {\map \exp {3 \mu + \dfrac {9 \sigma^2} 2} - 3 \map \exp {3 \mu + \dfrac {3 \sigma^2} 2} \paren {\map \exp {\sigma^2} - 1} - \map \exp {3 \mu + \dfrac {3 \sigma^2} 2} } {\map \exp {3 \mu + \dfrac {3 \sigma^2} 2} \paren {\map \exp {\sigma^2} - 1}^{\frac 3 2} }\)

Power of Power

\(\ds \)

\(=\)

\(\ds \frac {\map \exp {3 \mu + \dfrac {9 \sigma^2} 2} - 3 \map \exp {3 \mu + \dfrac {3 \sigma^2} 2} \map \exp {\sigma^2} + 3 \map \exp {3 \mu + \dfrac {3 \sigma^2} 2} - \map \exp {3 \mu + \dfrac {3 \sigma^2} 2} } {\map \exp {3 \mu + \dfrac {3 \sigma^2} 2} \paren {\map \exp {\sigma^2} - 1}^{\frac 3 2} }\)

Distributive Laws of Arithmetic

\(\ds \)

\(=\)

\(\ds \frac {\map \exp {3 \mu + \dfrac {9 \sigma^2} 2} - 3 \map \exp {3 \mu + \dfrac {5 \sigma^2} 2} + 2 \map \exp {3 \mu + \dfrac {3 \sigma^2} 2} } {\map \exp {3 \mu + \dfrac {3 \sigma^2} 2} \paren {\map \exp {\sigma^2} - 1}^{\frac 3 2} }\)

Product of Powers

\(\ds \)

\(=\)

\(\ds \frac {\map \exp {3 \mu + \dfrac {3 \sigma^2} 2} \paren {\map \exp {3 \sigma^2} - 3 \map \exp {\sigma^2} + 2} } {\map \exp {3 \mu + \dfrac {3 \sigma^2} 2} \paren {\map \exp {\sigma^2} - 1}^{\frac 3 2} }\)

factoring out $\map \exp {3 \mu + \dfrac {3 \sigma^2} 2}$

\(\ds \)

\(=\)

\(\ds \frac {\paren {\map \exp {3 \sigma^2} - 3 \map \exp {\sigma^2} + 2} } {\paren {\map \exp {\sigma^2} - 1}^{\frac 3 2} } \frac {\sqrt {\paren {\map \exp {\sigma^2} - 1} } } {\sqrt {\paren {\map \exp {\sigma^2} - 1} } }\)

canceling $\map \exp {3 \mu + \dfrac {3 \sigma^2} 2}$ and multiplying top and bottom by $\sqrt {\paren {\map \exp {\sigma^2} - 1} }$

\(\ds \)

\(=\)

\(\ds \frac {\paren {\map \exp {3 \sigma^2} - 3 \map \exp {\sigma^2} + 2} } {\paren {\map \exp {\sigma^2} - 1}^2} \sqrt {\paren {\map \exp {\sigma^2} - 1} }\)

rewriting

\(\ds \)

\(=\)

\(\ds \frac {\paren {\map \exp {3 \sigma^2} + \map \exp {\sigma^2} - 4 \map \exp {\sigma^2} + 2} } {\paren {\map \exp {\sigma^2} - 1}^2} \sqrt {\paren {\map \exp {\sigma^2} - 1} }\)

rewriting $-3 \map \exp {\sigma^2}$ as $\map \exp {\sigma^2} - 4 \map \exp {\sigma^2}$

\(\ds \)

\(=\)

\(\ds \frac {\paren {\map \exp {3 \sigma^2} - 2 \map \exp {2 \sigma^2} + \map \exp {\sigma^2} + 2 \map \exp {2 \sigma^2} - 4 \map \exp {\sigma^2} + 2} } {\paren {\map \exp {\sigma^2} - 1}^2} \sqrt {\paren {\map \exp {\sigma^2} - 1} }\)

adding and subtracting $2 \map \exp {2 \sigma^2}$

\(\ds \)

\(=\)

\(\ds \frac {\paren {\map \exp {\sigma^2} + 2} \paren {\map \exp {2 \sigma^2} - 2 \map \exp {\sigma^2} + 1} } {\paren {\map \exp {\sigma^2} - 1}^2} \sqrt {\paren {\map \exp {\sigma^2} - 1} }\)

rewriting

\(\ds \)

\(=\)

\(\ds \paren {\map \exp {\sigma^2} + 2} \sqrt {\paren {\map \exp {\sigma^2} - 1} }\)

canceling $\paren {\map \exp {\sigma^2} - 1}^2$

$\blacksquare$