Moment Generating Function of Logistic Distribution/Examples

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 $\size t < \dfrac 1 s$.

First Moment

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

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

Second Moment

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

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

Third Moment

Moment Generating Function of Logistic Distribution/Examples/Third Moment

Fourth Moment

Moment Generating Function of Logistic Distribution/Examples/Fourth Moment

Derivatives of Moment Generating Function of Logistic Distribution‎

The $n$th derivative of $M_X$ is given by:

$\ds {M_X}^{\paren n} = \map \exp {\mu t} \sum_{k \mathop = 0}^n \paren {-1}^k \dbinom n k \mu^{n - k} s^k \int_{\to 0}^{\to 1} \map {\ln^k} {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u$