Excess Kurtosis of Logistic Distribution

Theorem
Let $X$ be a continuous random variable which satisfies the logistic distribution:


 * $X \sim \map {\operatorname {Logistic} } {\mu, s}$

Then the excess kurtosis $\gamma_2$ of $X$ is equal to $\dfrac 6 5$.

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 Logistic Distribution we have:


 * $\mu = \mu$

By Variance of Logistic Distribution we have:


 * $\sigma = \dfrac {s \pi} {\sqrt 3}$

From Moment in terms of Moment Generating Function:
 * $\expect {X^n} = \map { {M_X}^{\paren n} } 0$

where $M_X$ is the moment generating function of $X$.

From Derivatives of Moment Generating Function of Logistic Distribution:


 * $\ds \map { {M_X}^{\paren 4} } 0 = \mu^4 \int_{\to 0}^{\to 1} \rd u - 4 \mu^3 s \int_{\to 0}^{\to 1} \map \ln {\dfrac {1 - u} u} \rd u + 6 \mu^2 s^2 \int_{\to 0}^{\to 1} \map {\ln^2} {\dfrac {1 - u} u} \rd u - 4 \mu s^3 \int_{\to 0}^{\to 1} \map {\ln^3} {\dfrac {1 - u} u} \rd u + s^4 \int_{\to 0}^{\to 1} \map {\ln^4} {\dfrac {1 - u} u} \rd u $

Therefore:

Hence: