Expectation 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 Moment in terms of Moment Generating Function:


 * $\expect X = \map { {M_X}'} 0$

From Moment Generating Function of Logistic Distribution: First Moment:


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

Hence setting $t = 0$: