Variance of Logistic Distribution/Proof 2

Proof
By Moment Generating Function of Logistic Distribution, the moment generating function of $X$ is given by:


 * $\ds \map {M_X} t = \map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u$

for $\size t < \dfrac 1 s$.

From Variance as Expectation of Square minus Square of Expectation:


 * $\ds \var X = \expect {X^2} - \paren {\expect X}^2$

From Expectation of Logistic Distribution, we have:


 * $\expect X = \mu$

From Moment Generating Function of Logistic Distribution: Second Moment:


 * $\ds \map { {M_X}''} t = \map \exp {\mu t} \paren {\mu^2 \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u - 2 s \mu \int_{\to 0}^{\to 1} \map \ln {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u + s^2 \int_{\to 0}^{\to 1} \map {\ln^2} {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u}$

Hence setting $t = 0$:

So: