The logistic function is a sigmoid function that maps the real numbers to the unit interval using the formula $f (x) = \frac{1}{1 + e^{- x}}$ .

$M$ is the upper bound of the function (in which case the function maps to the interval $(0, M)$ instead). When $M = 1$ , the logistic function is usually being used to calculate some probability or proportion of a total.
$x_{0}$ is the inflection point of the curve, or the value of $x$ where the function's growth stops speeding up and starts slowing down.
$α$ is a variable controlling the steepness of the curve.
$c$ is a scaling factor for the distance.

Applications

The logistic function is used to model growth rates of populations in an ecosystem with a limited carrying capacity.
The inverse logistic function (with $α = 2$ ) is used to convert a probability to log-odds form for use in Bayes' rule.
The logistic function (with $α = 10$ and $c = 1 / 400$ ) is used to calculate the expected probability of a player winning given a specific difference in rating in the Elo rating system.

Logistic function