Logistic distribution

Probability distribution
name =Logistic
type =density
pdf_

cdf_

parameters =$mu,$ location (real)
$s>0,$ scale (real)
support =$x\; in\; (-infty;\; +infty)!$
pdf =$frac\{e^\{-(x-mu)/s$ {sleft(1+e^{-(x-mu)/s} ight)^2}!
cdf =$frac\{1\}\{1+e^\{-(x-mu)/s!$
mean =$mu,$
median =$mu,$
mode =$mu,$
variance =$frac\{pi^2\}\{3\}\; s^2!$
skewness =$0,$
kurtosis =$6/5,$
entropy =$ln(s)+2,$
mgf =$e^\{mu,t\},mathrm\{B\}(1-s,t,;1+s,t)!$
for , Beta function
char =$e^\{i\; mu\; t\},mathrm\{B\}(1-ist,;1+ist),$
for
In probability theory and statistics, the logistic distribution is a continuous probability distribution.Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks.It resembles the normal distribution in shape but has heavier tails (higher kurtosis).

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:

:$F(x;\; mu,s)\; =\; frac\{1\}\{1+e^\{-(x-mu)/s\; !$::$=\; frac12\; +\; frac12\; ;operatorname\{tanh\}!left(frac\{x-mu\}\{2,s\}\; ight).$

Probability density function

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

:$f(x;\; mu,s)\; =\; frac\{e^\{-(x-mu)/s\; \{sleft(1+e^\{-(x-mu)/s\}\; ight)^2\}\; !$::$=frac\{1\}\{4,s\}\; ;operatorname\{sech\}^2!left(frac\{x-mu\}\{2,s\}\; ight).$

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:

:$F^\{-1\}(p;\; mu,s)\; =\; mu\; +\; s,lnleft(frac\{p\}\{1-p\}\; ight).$

Alternative parameterization

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

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

Applications

Both the United States Chess Federation and FIDE have switched their formulas for calculating chess ratings to the logistic distribution, see Elo rating system.

The logistic distribution and the S-shaped pattern that results from it have been extensively used in many different areas the most important of which include:

♦ Biology - to describe how species populations grow in competition [P. F. Verhulst, "Recherches mathématiques sur la loi d'accroissement de la population", "Nouveaux Mémoirs de l'Académie Royale des Sciences et des Belles-Lettres de Bruxelles", vol. 18 (1845); Alfred J. Lotka, "Elements of Physical Biology", (Baltimore, MD: Williams & Wilkins Co., 1925).]

♦ Epidemiology - to describe the spreading of epidemics [Theodore Modis, "Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future", Simon & Schuster, New York, 1992, pp 97-105.]

♦ Psychology - to describe learning [Theodore Modis, "Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future", Simon & Schuster, New York, 1992, Chapter 2.]

♦ Technology - to describe how new technologies diffuse and substitute for each other [J. C. Fisher and R. H. Pry , "A Simple Substitution Model of Technological Change", "Technological Forecasting & Social Change", vol. 3, no. 1 (1971).]

♦ Market - the diffusion of new-product sales [Theodore Modis, "Conquering Uncertainty", McGraw-Hill, New York, 1998, Chapter 1.]

♦ Energy - the diffusion and substitution of primary energy sources [Cesare Marchetti, "Primary Energy Substitution Models: On the Interaction between Energy and Society", "Technological Forecasting & Social Change", vol. 10, (1977).]

Related distributions

If log("X") has a logistic distribution then "X" has a log-logistic distribution and "X" – "a" has a shifted log-logistic distribution.

Derivations

Expected Value

:$E\; [X]\; =int\_\{-infty\}^\{infty\}\; \{frac\{xe^\{-(x-mu)/s\; \{sleft(1+e^\{-(x-mu)/s\}\; ight)^2\; !\; dx\; =\; int\_\{-infty\}^\{infty\}\; frac\{x\}\{4,s\}\; ;operatorname\{sech\}^2!left(frac\{x-mu\}\{2,s\}\; ight)dx$

:Subsitute: $u=frac\{(x-mu)\}\{2s\},\; du=frac\{1\}\{2s\}\; dx$

:$E\; [X]\; =int\_\{-infty\}^\{infty\}\; frac\{2,s,u+mu\}\{2\}\; ;operatorname\{sech\}^2!left(u\; ight)du$

:$E\; [X]\; =sint\_\{-infty\}^\{infty\}\; u\; ;operatorname\{sech\}^2!left(u\; ight)du\; +\; frac\{mu\}\{2\}\; int\_\{-infty\}^\{infty\}\; ;operatorname\{sech\}^2!left(u\; ight)du$

:Note the odd function: $int\_\{-infty\}^\{infty\}\; u\; ;operatorname\{sech\}^2!left(u\; ight)du\; =\; 0$

:$E\; [X]\; =frac\{mu\}\{2\}\; int\_\{-infty\}^\{infty\}\; ;operatorname\{sech\}^2!left(u\; ight)du\; =\; frac\{mu\}\{2\},2\; =\; mu$

See also

* logistic regression
* sigmoid function

Notes

References

* cite book
first = Balakrishnan
last = N.
year = 1992
title = Handbook of the Logistic Distribution
publisher = Marcel Dekker, New York
id = ISBN 0-8247-8587-8
* cite book
author = Johnson, N. L., Kotz, S., Balakrishnan N.
year = 1995
title = Continuous Univariate Distributions
others = Vol. 2
edition = 2nd Ed.
id = ISBN 0-471-58494-0