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 for Derivatives, Chain Rule for Derivatives Derivative of Power of Constant and Derivative of Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \exp {\mu t} \paren {\mu \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u - s \int_{\to 0}^{\to 1} \map \ln {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u}\) |
$\blacksquare$