Moment Generating Function of Logistic Distribution/Examples/Second Moment

Examples of Use of Moment Generating Function of Logistic Distribution
Let $X$ be a continuous random variable which satisfies the logistic distribution:


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

for some $\mu \in \R, s \in \R_{> 0}$.

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

The second moment generating function of $X$ is given by:


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

Proof
We have: