Logistics Distribution

A random variable $X$ is said to have a logistics distribution with parameters $\mu$ and $v$ if its cumulative density function is of the form \begin{align} F_{X}(x) = \frac{e^{(x-\mu)/ v}}{1 + e^{(x-\mu)/ v}}, \quad \text{ $\forall - \infty < x < \infty$} \end{align} Differentiating to get the density function \begin{align} f_{x}(x) = \frac{e^{(x-\mu)/v}}{v(1 + e^{(x-\mu)/v})^{2}}, \quad \text{ $\forall -\infty < x < \infty$} \end{align}

Mean

\begin{align} E[X] &= \mu\newline v &= \text{dispersion parameter} \end{align}