Logistics Distribution¶
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
\[
\begin{aligned}
F_{X}(x) = \frac{e^{(x-\mu)/ v}}{1 + e^{(x-\mu)/ v}}, \quad \text{ $\forall - \infty < x < \infty$}
\end{aligned}
\]
Differentiating to get the density function
\[
\begin{aligned}
f_{x}(x) = \frac{e^{(x-\mu)/v}}{v(1 + e^{(x-\mu)/v})^{2}}, \quad \text{ $\forall -\infty < x < \infty$}
\end{aligned}
\]
Mean¶
\[
\begin{aligned}
E[X] &= \mu\newline
v &= \text{dispersion parameter}
\end{aligned}
\]