Moment Generating Function of Logistic Distribution

Theorem
Let $X$ be a continuous random variable which satisfies the logistic distribution:


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

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

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


 * $\map {M_X} t = \begin {cases} \map \exp {\mu t} \map \Beta {\paren {1 - s t}, \paren {1 + s t} } & \size t < \dfrac 1 s  \\ \text {does not exist} & \size t \ge \dfrac 1 s \end {cases}$

where $\Beta$ denotes the beta function

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


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

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:


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

let:

and also:

Then:

Note that by definition of the beta function, $\Beta: \C \times \C \to \C$ is defined only for $\map \Re x, \map \Re y > 0$:

Therefore:
 * $1 - s t > 0 \leadsto t < \dfrac 1 s$

and:
 * $1 + s t > 0 \leadsto t > -\dfrac 1 s$

Therefore, the moment generating function is defined only when $\size t < \dfrac 1 s$.