Cumulative Distribution Function of Logistic Distribution

Theorem
Let $X$ be a continuous random variable with the logistic distribution.

Then the cumulative distribution function of $X$ is:
 * $\map {F_X} x = \dfrac 1 {1 + \map \exp {- \dfrac {x - \mu} s} }$

Proof
The derivative of $F_X$ is:

By the Fundamental Theorem of Calculus:
 * $\ds \int_a^b \map {f_X} x \rd x = \bigintlimits {\map {F_X} x} {x \mathop = a} {x \mathop = b}$

Therefore:

Hence the result.