Variance of Logistic Distribution/Proof 3

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 Variance as Expectation of Square minus Square of Expectation:


 * $\ds \var X = \int_{-\infty}^\infty x^2 \, \map {f_X} x \rd x - \paren {\expect X}^2$

So:

From Sum of Infinite Geometric Sequence, for $x > \mu$, we have:


 * $\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \map \exp {-\dfrac {\paren {x - \mu} } s}^n = \dfrac 1 {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }$

Taking the derivative of both sides, we have:

Therefore: