Excess Kurtosis 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 excess kurtosis $\gamma_2$ of $X$ is given by:


 * $\gamma_2 = \exp {\paren {4\sigma^2 } } + 2 \exp {\paren {3\sigma^2  } } + 3 \exp {\paren {2\sigma^2  } } - 6 $

Proof
From Kurtosis in terms of Non-Central Moments, we have:


 * $\gamma_2 = \dfrac {\expect {X^4} - 4 \mu \expect {X^3} + 6 \mu^2 \expect {X^2} - 3 \mu^4} {\sigma^4} - 3$

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:


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

Hence: