Moment Generating Function of Logistic Distribution

Theorem
Let $X \sim \map {\operatorname {Logistic} } {\mu, {\sigma} }$ for some $\mu \in \R, \sigma \in \R_{> 0}$.

Then the moment generating function $M_X$ of $X$ is given by:


 * $\map {M_X} t = \map \exp {\mu t} \map \Beta {\paren {1 - \sigma t}, \paren {1 + \sigma t} }$

Proof
From the definition of the logistic distribution, $X$ has probability density function:


 * $\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } \sigma} } {\sigma \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } \sigma} }^2}$

From the definition of a moment generating function:


 * $\ds \map {M_X} t = \expect { e^{t X} } = \int_{-\infty}^\infty e^{t x} \map {f_X} x \rd x$

So:


 * $\ds \map {M_X} t = \frac 1 {\sigma } \int_{-\infty}^\infty \dfrac {\map \exp {tx} \map \exp {-\dfrac {\paren {x - \mu} } \sigma} } {\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } \sigma} }^2} \rd x$

let:

and also:

Then:

Now let:

and also:

Then: