Editing Logistic distribution (section)

== Applications ==
The logistic distribution—and the S-shaped pattern of its [[cumulative distribution function]] (the [[logistic function]]) and [[quantile function]] (the [[logit function]])—have been extensively used in many different areas.

=== Logistic regression ===
One of the most common applications is in [[logistic regression]], which is used for modeling [[categorical variable|categorical]] [[dependent variable]]s (e.g., yes-no choices or a choice of 3 or 4 possibilities), much as standard [[linear regression]] is used for modeling [[continuous variable]]s (e.g., income or population). Specifically, logistic regression models can be phrased as [[latent variable]] models with [[error variable]]s following a logistic distribution. This phrasing is common in the theory of [[discrete choice]] models, where the logistic distribution plays the same role in logistic regression as the [[normal distribution]] does in [[probit regression]]. Indeed, the logistic and normal distributions have a quite similar shape. However, the logistic distribution has [[heavy-tailed distribution|heavier tails]], which often increases the [[robust statistics|robustness]] of analyses based on it compared with using the normal distribution.

=== Physics ===
The PDF of this distribution has the same functional form as the derivative of the [[Fermi function]]. In the theory of electron properties in semiconductors and metals, this derivative sets the relative weight of the various electron energies in their contributions to electron transport. Those energy levels whose energies are closest to the distribution's "mean" ([[Fermi level]]) dominate processes such as electronic conduction, with some smearing induced by temperature.<ref>{{Cite book | isbn = 9780521484916 | title = The Physics of Low-dimensional Semiconductors: An Introduction | last1 = Davies | first1 = John H. | year = 1998 | publisher = Cambridge University Press  }}</ref>{{rp|34}} However the pertinent ''probability'' distribution in [[Fermi–Dirac statistics]] is actually a simple [[Bernoulli distribution]], with the probability factor given by the Fermi function.

The logistic distribution arises as limit distribution of a finite-velocity damped random motion described by a telegraph process in which the random times between consecutive velocity changes have independent exponential distributions with linearly increasing parameters.<ref>A. Di Crescenzo, B. Martinucci (2010) "A damped telegraph random process with logistic stationary distribution", ''[[Applied Probability Trust|J. Appl. Prob.]]'', vol. 47, pp. 84–96.</ref>

=== Hydrology ===
[[File:FitLogisticdistr.tif|thumb|250px|Fitted cumulative logistic distribution to October rainfalls using [[CumFreq]], see also [[Distribution fitting]] ]]
In [[hydrology]] the distribution of long duration river discharge and rainfall (e.g., monthly and yearly totals, consisting of the sum of 30 respectively 360 daily values) is often thought to be almost normal according to the [[central limit theorem]].<ref>{{cite book|editor-last=Ritzema|editor-first=H.P.|title=Frequency and Regression Analysis|year=1994|publisher=Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands|pages=[https://archive.org/details/drainageprincipl0000unse/page/175 175–224]|url=https://archive.org/details/drainageprincipl0000unse/page/175|isbn=90-70754-33-9}}</ref> The [[normal distribution]], however, needs a numeric approximation. As the logistic distribution, which can be solved analytically, is similar to the normal distribution, it can be used instead. The blue picture illustrates an example of fitting the logistic distribution to ranked October rainfalls—that are almost normally distributed—and it shows the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].

=== Chess ratings ===
The [[United States Chess Federation]] and FIDE have switched its formula for calculating chess ratings from the normal distribution to the logistic distribution; see the article on [[Elo rating system]] (itself based on the normal distribution).