Moment Generating Function of Logistic Distribution/Examples/First Moment

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

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


Proof

We have:

\(\ds \map { {M_X}'} t\) \(=\) \(\ds \map {\frac \d {\d t} } {\map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u}\) Moment Generating Function of Logistic Distribution
\(\ds \) \(=\) \(\ds \mu \map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u + \map \exp {\mu t} \int_{\to 0}^{\to 1} -s \map \ln {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u\) Product Rule, Chain Rule for Derivatives Derivative of Power of Constant and Derivative of Exponential Function
\(\ds \) \(=\) \(\ds \map \exp {\mu t} \paren {\mu \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u - s \int_{\to 0}^{\to 1} \map \ln {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u}\)

$\blacksquare$