{s\left(1+e^{-(x-\mu)/s}\right)^2}\!| cdf =$\backslash frac\{1\}\{1+e^\{-(x-\backslash mu)/s\}\}\backslash !$| mean =$\backslash mu\backslash ,$| median =$\backslash mu\backslash ,$| mode =$\backslash mu\backslash ,$| variance =$\backslash frac\{\backslash pi^2\}\{3\}\; s^2\backslash !$| skewness =$0\backslash ,$| kurtosis =$6/5\backslash ,$| entropy =$\backslash ln(s)+2\backslash ,$| mgf =$e^\{m\backslash ,t\}\backslash ,\backslash mathrm\{B\}(1-s\backslash ,t,\backslash ;1+s\backslash ,t)\backslash !$
for | char =$e^\{imt\}\backslash ,\backslash mathrm\{B\}(1-ist,\backslash ;1+ist)\backslash ,$
for | }} In probability theory and statistics, the logistic distribution is a continuous probability distribution. It is derived from the work of Pierre François Verhulst (1804–1849), Professor of Analysis at the Belgian Military College, in modelling the growth of population in Belgium in the early 1800's. Verhulst's description of the growth of population follows the cumulative distribution function of the logistic distribution (also known as the "logistic ogive"). Population grows geometrically for small populations which have more resources than they need, then becomes constant as the resources become fully utilized, then becomes very slow as the population demand for resources exceeds the supply. The logistic distribution is closely related to the logistic function and the logistic equation which also follow from the work of Verhulst.

This distribution has longer tails than the normal distribution and a higher kurtosis of 1.2 (compared with 0 for the normal distribution).

Specification

Cumulative distribution function

The logistic distribution receives its name from its cumulative distribution function (cdf), which is an instance of the family of logistic functions:

$F(x;\; \backslash mu,s)\; =\; \backslash frac\{1\}\{1+e^\{-(x-\backslash mu)/s\}\}\; \backslash !$

$=\; \backslash frac12\; +\; \backslash frac12\; \backslash ;\backslash operatorname\{tanh\}\backslash !\backslash left(\backslash frac\{x-\backslash mu\}\{2\backslash ,s\}\backslash right)\; \backslash !$

Probability density function

The probability density function (pdf) of the logistic distribution is given by:

$f(x;\; \backslash mu,s)\; =\; \backslash frac\{e^\{-(x-\backslash mu)/s\}\}\; \{s\backslash left(1+e^\{-(x-\backslash mu)/s\}\backslash right)^2\}\; \backslash !$

$=\backslash frac\{1\}\{4\backslash ,s\}\; \backslash ;\backslash operatorname\{sech\}^2\backslash !\backslash left(\backslash frac\{x-\backslash mu\}\{2\backslash ,s\}\backslash right)\; \backslash !$

Because the pdf can be expressed in terms of the square of the hyperbolic secant function "sech", it is sometimes referred to as the sech-square(d) distribution.

See also: hyperbolic secant distribution

Quantile function

The inverse cumulative distribution function of the logistic distribution is $F^\{-1\}$, a generalization of the logit function, defined as follows:

$F^\{-1\}(p;\; \backslash mu,s)\; =\; \backslash mu\; +\; s\backslash ,\backslash ln\backslash left(\backslash frac\{p\}\{1-p\}\backslash right)\; \backslash !$

Alternative parameterization

An alternative parameterization of the logistic distribution can be derived using the substitution $\backslash sigma^2\; =\; \backslash pi^2\backslash ,s^2/3$. This yields the following density function:

$g(x;\backslash mu,\backslash sigma)\; =\; f(x;\backslash mu,\backslash sigma\backslash sqrt\{3\}/\backslash pi)\; =\; \backslash frac\{\backslash pi\}\{\backslash sigma\backslash ,4\backslash sqrt\{3\}\}\; \backslash ,\backslash operatorname\{sech\}^2\backslash !\backslash left(\backslash frac\{\backslash pi\}\{2\; \backslash sqrt\{3\}\}\; \backslash ,\backslash frac\{x-\backslash mu\}\{\backslash sigma\}\backslash right)\; \backslash !$

References

See also

• logistic regression
• sigmoid function

Continuous distributions

Logistische Verteilung