We recommend reading this documentation on
https://alexpghayes.github.io/distributions3/, where the math
will render with additional detail and much greater clarity.
In the following, let \(X\) be a Logistic random variable with
location = \(\mu\) and scale = \(s\).
Support: \(R\), the set of all real numbers
Mean: \(\mu\)
Variance: \(s^2 \pi^2 / 3\)
Probability density function (p.d.f):
$$
f(x) = \frac{e^{-(\frac{x - \mu}{s})}}{s [1 + \exp(-(\frac{x - \mu}{s})) ]^2}
$$
Cumulative distribution function (c.d.f):
$$
F(t) = \frac{1}{1 + e^{-(\frac{t - \mu}{s})}}
$$
Moment generating function (m.g.f):
$$
E(e^{tX}) = e^{\mu t} \beta(1 - st, 1 + st)
$$
where \(\beta(x, y)\) is the Beta function.