Moment Generating Function of Logistic Distribution

From ProofWiki
Jump to navigation Jump to search

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:

\(\ds u\) \(=\) \(\ds \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}\) Integration by Substitution
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds -\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-2} \paren {-\frac 1 s \map \exp {-\dfrac {\paren {x - \mu} } s} }\) Power Rule for Derivatives, Chain Rule for Derivatives and Derivative of Exponential Function: Corollary 1
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 u - 1\) \(=\) \(\ds \paren {\map \exp {-\dfrac {\paren {x - \mu} } s} }\)
\(\ds \leadsto \ \ \) \(\ds \map \exp {-\dfrac \mu s} \paren {\dfrac 1 u - 1}\) \(=\) \(\ds \map \exp {-\dfrac \mu s} \paren {\map \exp {-\dfrac {\paren {x - \mu} } s} }\) multiplying both sides by $\map \exp {-\dfrac \mu s}$
\(\ds \leadsto \ \ \) \(\ds \map \exp {-\dfrac \mu s} \paren {\dfrac 1 u - 1}\) \(=\) \(\ds \paren {\map \exp {-\dfrac x s} }\)
\(\ds \leadsto \ \ \) \(\ds \paren {\map \exp {-\dfrac \mu s} \paren {\dfrac 1 u - 1} }^{-s t}\) \(=\) \(\ds \paren {\map \exp {-\dfrac x s} }^{-s t}\) raising both sides to the $-s t$ power
\(\ds \leadsto \ \ \) \(\ds \map \exp {\mu t} \paren {\dfrac {1 - u} u}^{-s t}\) \(=\) \(\ds \map \exp {x t}\)


and also:

\(\ds \lim_{x \mathop \to -\infty} \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}\) \(=\) \(\ds 0\)
\(\ds \lim_{x \mathop \to \infty} \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}\) \(=\) \(\ds 1\)


Then:

\(\ds \map {M_X} t\) \(=\) \(\ds \map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u\)
\(\ds \) \(=\) \(\ds \map \exp {\mu t} \int_{\to 0}^{\to 1} \paren {1 - u }^{-s t} u^{s t} \rd u\)
\(\ds \) \(=\) \(\ds \map \exp {\mu t} \map \Beta {\paren {1 - s t}, \paren {1 + s t} }\) Definition of Beta Function and Commutativity of Parameters of Beta Function


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 $\size t < \dfrac 1 s$.

$\blacksquare$


Examples

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