Theorem: Skewness of Log Normal Distribution

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:
{{begin-eqn}}
{{eqn | l = \gamma_1
      | r = \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}
}}
{{eqn | r = \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} }
      | c = Power of Power
}}
{{eqn | r = \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} }
      | c = Distributive Laws of Arithmetic
}}
{{eqn | r = \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} }
      | c = Product of Powers
}}
{{eqn | r = \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} }
      | c = factoring out $\map \exp {3 \mu + \dfrac {3 \sigma^2} 2}$
}}
{{eqn | r = \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} } }
      | c = canceling $\map \exp {3 \mu + \dfrac {3 \sigma^2} 2}$ and multiplying top and bottom by $\sqrt {\paren {\map \exp {\sigma^2} - 1} }$
}}
{{eqn | r = \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} } 
      | c = rewriting
}}
{{eqn | r = \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} }
      | c = rewriting $-3 \map \exp {\sigma^2}$ as $\map \exp {\sigma^2} - 4 \map \exp {\sigma^2}$
}}
{{eqn | r = \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} }
      | c = adding and subtracting $2 \map \exp {2 \sigma^2}$
}}
{{eqn | r = \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} }
      | c = rewriting
}}
{{eqn | r = \paren {\map \exp {\sigma^2} + 2} \sqrt {\paren {\map \exp {\sigma^2} - 1} } 
      | c = canceling $\paren {\map \exp {\sigma^2} - 1}^2$
}}
{{end-eqn}}
{{qed}}
Category:Skewness
Category:Log Normal Distribution
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