Expectation of Logistic Distribution/Proof 1

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 the expected value of a continuous random variable:


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

So:


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

let:

and also:

Then: