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| {{Probability distribution|
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| name =Logistic|
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| type =density|
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| pdf_image =[[File:Logisticpdfunction.svg|320px|Standard logistic PDF]]|
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| cdf_image =[[File:Logistic cdf.svg|320px|Standard logistic CDF]]|
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| parameters =μ [[location parameter|location]] ([[real number|real]])<br />''s'' > 0 [[scale parameter|scale]] (real)|
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| support =''x'' ∈ (−∞, ∞)|
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| pdf =<math>\frac{e^{-\frac{x-\mu}{s}}} {s\left(1+e^{-\frac{x-\mu}{s}}\right)^2}\!</math>|
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| cdf =<math>\frac{1}{1+e^{-\frac{x-\mu}{s}}}\!</math>|
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| mean =<math>\mu</math>|
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| median =<math>\mu</math>|
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| mode =<math>\mu</math>|
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| variance =<math>\tfrac{s^2 \pi^2}{3}</math>|
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| skewness =<math>0</math>|
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| kurtosis =<math>1.2</math>|
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| entropy =<math>ln\left(s\right) + 2 = ln\left(\sigma\right) + 1.404576+,</math><br/>where ''σ'' is the standard deviation.|
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| mgf =<math>e^{\mu t}\,\mathrm{B}(1-st, 1+st)</math><br />for ''st'' ∈ (−1, 1) [[Beta function]]|
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| char =<math>e^{it\mu}\frac{\pi st}{\sinh(\pi st)}</math>}}
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| 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 network]]s. It resembles the [[normal distribution]] in shape but has heavier tails (higher [[kurtosis]]). The [[Tukey lambda distribution]] can be considered a generalization of the logistic distribution since it adds a shape parameter, ''λ''. The logistic distribution is obtained by setting ''λ'' to zero.
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| == Specification ==
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| === Probability density function ===
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| The [[probability density function]] (pdf) of the logistic distribution is given by:
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| :<math>f(x; \mu,s) = \frac{e^{-\frac{x-\mu}{s}}} {s\left(1+e^{-\frac{x-\mu}{s}}\right)^2} =\frac{1}{4s} \operatorname{sech}^2\!\left(\frac{x-\mu}{2s}\right).</math>
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| Because the pdf can be expressed in terms of the square of the [[hyperbolic function|hyperbolic secant function]] "sech", it is sometimes referred to as the ''sech-square(d) distribution''.<ref>Johnson, Kotz & Balakrishnan (1995, p.116).</ref>
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| :''See also:'' [[hyperbolic secant distribution]]
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| === Cumulative distribution function ===
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| The logistic distribution receives its name from its [[cumulative distribution function]] (cdf), which is an instance of the family of logistic functions. The cumulative distribution function of the logistic distribution is also a scaled version of the [[Hyperbolic function|hyperbolic tangent]].
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| :<math>F(x; \mu, s) = \frac{1}{1+e^{-\frac{x-\mu}{s}}} = \frac12 + \frac12 \;\operatorname{tanh}\!\left(\frac{x-\mu}{2s}\right).</math>
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| In this equation, ''x'' is the [[random variable]], μ is the [[mean]], and ''s'' is a scale parameter proportional to the [[standard deviation]].
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| === Quantile function ===
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| The [[inverse function|inverse]] cumulative distribution function (quantile function) of the logistic distribution is a generalization of the [[logit]] function. Its derivative is called the quantile density function. They are defined as follows:
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| :<math>Q(p;\mu,s) = \mu + s\,\ln\left(\frac{p}{1-p}\right).</math> | |
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| :<math>Q'(p;s) = \frac{s}{p(1-p)}.</math>
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| == Alternative parameterization ==
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| An alternative parameterization of the logistic distribution can be derived by expressing the scale parameter, <math>s</math>, in terms of the standard deviation, <math>\sigma</math>, using the substitution <math>s\,=\,q\,\sigma</math>, where <math>q\,=\,\sqrt{3}/{\pi}\,=\,0.551328895+</math>. The alternative forms of the above functions are reasonably straightforward.
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| == Applications ==
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| 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. 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.
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| Other applications:
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| [[File:FitLogisticdistr.tif|thumb|250px|Fitted cumulative logistic distribution to October rainfalls using [[CumFreq]], see also [[Distribution fitting]] ]]
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| *Hydrology - 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|last=Ritzema (ed.)|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=175–224|url=http://www.waterlog.info/pdf/freqtxt.pdf|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]].
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| *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 isbn|9780521484916}}</ref>{{rp|34}} Note however that the pertinent ''probability'' distribution in [[Fermi–Dirac statistics]] is actually a simple [[Bernoulli distribution]], with the probability factor given by the Fermi function.
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| Both the [[United States Chess Federation]] and [[FIDE]] have switched their formulas for calculating chess ratings from the normal distribution to the logistic distribution; see [[Elo rating system]].
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| 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, p. 84-96.</ref>
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| == Related distributions ==
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| * Logistic distribution mimics the [[Sech distribution]].
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| * If ''X'' ~ Logistic(μ, β) then ''kX'' + loc ~ Logistic(''k''μ+loc, ''k''β).
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| * If ''X'' ~ [[Uniform distribution (continuous)|''U''(0, 1)]] then μ + β(log(''X'')−log(1−''X'')) ~ Logistic(μ, β).
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| * If ''X'', ''Y'' ~ [[Gumbel distribution|Gumbel(α, β)]] then ''X''−''Y'' ~ Logistic(0, β).
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| * If ''X'', ''Y'' ~ [[Generalized extreme value distribution|GEV(α, β, 0)]] then ''X''−''Y'' ~ Logistic(0, β).
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| * If ''X'' ~ Gumbel(α, β) and ''Y'' ~ GEV(α, β, 0) then ''X''+''Y'' ~ Logistic(2α, β).
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| * If log(''X'') ~ Logistic then ''X'' ~ [[log-logistic distribution|LogLogistic]] and ''X''−''a'' ~ [[shifted log-logistic distribution|ShiftedLogLogistic]].
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| * If ''X'' ~ [[Exponential distribution|Exponential(1)]] then
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| ::<math>\mu-\beta\log \left(\frac{e^{-X}}{1-e^{-X}} \right) \sim \mathrm{Logistic}(\mu,\beta). </math> | |
| * If ''X'', ''Y'' ~ Exponential(1) then
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| ::<math>\mu-\beta\log\left(\frac{X}{Y}\right) \sim \mathrm{Logistic}(\mu,\beta).</math>
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| == Derivations ==
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| === Higher order moments ===
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| The ''n''-th order central moment can be expressed in terms of the quantile function:
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| :<math>\operatorname{E}[(X-\mu)^n] = \int_{-\infty}^\infty (x-\mu)^n dF(x) = \int_0^1\big(Q(p)-\mu\big)^n dp = s^n \int_0^1 \left[\ln\!\left(\frac{p}{1-p}\right)\right]^n dp. </math>
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| This integral is well-known<ref>{{OEIS2C|A001896}}</ref> and can be expressed in terms of [[Bernoulli number]]s:
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| : <math> \operatorname{E}[(X-\mu)^n] = s^n\pi^n(2^n-2)\cdot|B_n|.</math>
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| == See also ==
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| * [[Generalized logistic distribution]]
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| * [[Tukey lambda distribution]]
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| * [[Logistic regression]]
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| * [[Log-logistic distribution]]
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| * [[Sigmoid function]]
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| == Notes ==
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| <references/>
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| == References ==
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| * {{Cite journal
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| | author = John S. deCani and Robert A. Stine
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| | year = 1986
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| | title = A note on deriving the information matrix for a logistic distribution
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| | journal = The American Statistician
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| | volume = 40
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| | pages = 220–222
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| | publisher = American Statistical Association
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| }}
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| * {{Cite book
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| | first = Balakrishnan
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| | last = N.
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| | year = 1992
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| | title = Handbook of the Logistic Distribution
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| | publisher = Marcel Dekker, New York
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| | isbn = 0-8247-8587-8
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| }}
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| * {{cite book
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| | author = Johnson, N. L., Kotz, S., Balakrishnan N.
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| | year = 1995
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| | title = Continuous Univariate Distributions
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| | others = Vol. 2
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| | edition = 2nd
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| | isbn = 0-471-58494-0 }}
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| *Modis, Theodore (1992) ''Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future'', Simon & Schuster, New York. ISBN 0-671-75917-5
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| == External links ==
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| {{commons category}}
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| {{ProbDistributions|continuous-infinite}}
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| {{DEFAULTSORT:Logistic Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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