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  text: '[summary: The logistic function is a [-sigmoid] function that maps the [4bc real numbers] to the unit interval $(0, 1)$ using the formula $\\displaystyle f(x) = \\frac{1}{1 + e^{-x}}$ or variants of this formula.]\n\nThe logistic function is a [-sigmoid] function that maps the [4bc real numbers] to the unit interval $(0, 1)$ using the formula $\\displaystyle f(x) = \\frac{1}{1 + e^{-x}}$.\n\nMore generally, there exists a [family_of_functions family] of logistic functions that can be written as $\\displaystyle f(x) = \\frac{M}{1 + \\alpha^{c(x_0 - x)}}$, where:\n\n* $M$ 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 [-1rf] or [-4w3] of a total.\n\n* $x_0$ is the inflection point of the curve, or the value of $x$ where the function's growth stops speeding up and starts slowing down.\n\n* $\\alpha$ is a variable controlling the steepness of the curve.\n\n* $c$ is a scaling factor for the distance.\n\n## Applications\n\n* The logistic function is used to model growth rates of populations in an ecosystem with a limited carrying capacity.\n\n* The inverse logistic function (with $\\alpha = 2$) is used to convert a probability to log-odds form for use in [1lz].\n\n* The logistic function (with $\\alpha = 10$ 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.',
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