Logistic distribution#Cumulative distribution function

The logistic distribution receives its name from its cumulative distribution function, 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 tangent.

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

In this equation {{math|μ}} is the mean, and {{math|s}} is a scale parameter proportional to the standard deviation.

The probability density function is the partial derivative of the cumulative distribution function:

: $$

\begin{align}

f(x; \mu,s) & = \frac{\partial F(x; \mu, s)}{\partial x} = \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2} \\[4pt]

& =\frac{1}{s\left(e^{(x-\mu)/(2s)}+e^{-(x-\mu)/(2s)}\right)^2} \\[4pt]

& =\frac{1}{4s} \operatorname{sech}^2\left(\frac{x-\mu}{2s}\right).

\end{align}

When the location parameter {{math|μ}} is 0 and the scale parameter {{math|s}} is 1, then the probability density function of the logistic distribution is given by

: $$

\begin{align}

f(x; 0,1) & = \frac{e^{-x}}{(1+e^{-x})^2} \\[4pt]

& = \frac 1 {(e^{x/2} + e^{-x/2})^2} \\[5pt]

& = \frac 1 4 \operatorname{sech}^2 \left(\frac x 2 \right).

\end{align}

Because this function 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.Johnson, Kotz & Balakrishnan (1995, p.116). (See also: hyperbolic secant distribution).

The 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:

:$Q(p;\backslash mu,s)\; =\; \backslash mu\; +\; s\; \backslash ln\backslash left(\backslash frac\{p\}\{1-p\}\backslash right).$

:$Q\text{'}(p;s)\; =\; \backslash frac\{s\}\{p(1-p)\}.$

An alternative parameterization of the logistic distribution can be derived by expressing the scale parameter, $s$, in terms of the standard deviation, $\backslash sigma$, using the substitution $s\backslash ,=\backslash ,q\backslash ,\backslash sigma$, where $q\backslash ,=\backslash ,\backslash sqrt\{3\}/\{\backslash pi\}\backslash ,=\backslash ,0.551328895\backslash ldots$. The alternative forms of the above functions are reasonably straightforward.

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 dependent variables (e.g., yes-no choices or a choice of 3 or 4 possibilities), much as standard linear regression is used for modeling continuous variables (e.g., income or population). Specifically, logistic regression models can be phrased as latent variable models with error variables 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 heavier tails, which often increases the robustness of analyses based on it compared with using the normal distribution.

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.{{Cite book | isbn = 9780521484916 | title = The Physics of Low-dimensional Semiconductors: An Introduction | last1 = Davies | first1 = John H. | year = 1998 | publisher = Cambridge University Press }}{{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.A. Di Crescenzo, B. Martinucci (2010) "A damped telegraph random process with logistic stationary distribution", J. Appl. Prob., vol. 47, pp. 84–96.

File:FitLogisticdistr.tif, 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.{{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}} 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 positions as part of the cumulative frequency analysis.

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).

• Logistic distribution mimics the sech distribution; they are different cases of the Champernowne distribution.
• If $X\; \backslash sim\; \backslash mathrm\{Logistic\}(\backslash mu,\; s)$ then $kX\; +\; \backslash ell\; \backslash sim\; \backslash mathrm\{Logistic\}(k\backslash mu\; +\; \backslash ell,\; |k|s)$.
• If $X\; \backslash sim$ U(0, 1) then $\backslash mu\; +\; s\; \backslash cdot\; \backslash text\{logit\}(X)\; \backslash sim\; \backslash mathrm\{Logistic\}(\backslash mu,\; s)$, where $\backslash text\{logit\}(X)=\backslash log\; X-\backslash log(1-X)$ is the logit function.
• If $X\; \backslash sim\; \backslash mathrm\{Gumbel\}(\backslash mu\_X,\; \backslash beta)$ and $Y\; \backslash sim\; \backslash mathrm\{Gumbel\}(\backslash mu\_Y,\; \backslash beta)$ independently then $X-Y\; \backslash sim\; \backslash mathrm\{Logistic\}(\backslash mu\_X-\backslash mu\_Y,\backslash beta)\; \backslash ,$.
• If $X$ and $Y\; \backslash sim\; \backslash mathrm\{Gumbel\}(\backslash mu,\; \backslash beta)$ then $X+Y\; \backslash nsim\; \backslash mathrm\{Logistic\}(2\; \backslash mu,\backslash beta)\; \backslash ,$ (The sum is not a logistic distribution). $E(X+Y)\; =\; 2\backslash mu+2\backslash beta\backslash gamma\; \backslash neq\; 2\backslash mu\; =\; E\backslash left(\backslash mathrm\{Logistic\}(2\; \backslash mu,\backslash beta)\; \backslash right)$.
• If X ~ Logistic(μ, s) then exp(X) ~ LogLogistic$\backslash left(\; \backslash alpha\; =\; e^\backslash mu,\; \backslash beta\; =\; \backslash frac\; 1\; s\; \backslash right)$, and exp(X) + γ ~ shifted log-logistic$\backslash left(\; \backslash alpha\; =\; e^\backslash mu,\; \backslash beta\; =\; \backslash frac\; 1\; s,\; \backslash gamma\; \backslash right)$.
• If X ~ Exponential(1) then

::$\backslash mu+s\backslash log(e^X\; -1)\; \backslash sim\; \backslash operatorname\{Logistic\}(\backslash mu,s).$

• If X, Y ~ Exponential(λ) independently then

::$\backslash mu+s\backslash log\backslash left(\backslash frac\; X\; Y\; \backslash right)\; \backslash sim\; \backslash operatorname\{Logistic\}(\backslash mu,s).$

• The metalog distribution is generalization of the logistic distribution, in which power series expansions in terms of $p$ are substituted for logistic parameters $\backslash mu$ and $\backslash sigma$. The resulting metalog quantile function is highly shape flexible, has a simple closed form, and can be fit to data with linear least squares.

The nth-order central moment can be expressed in terms of the quantile function:

: $$

\begin{align}

\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.

\end{align}

This integral is well-known{{OEIS2C|A001896}} and can be expressed in terms of Bernoulli numbers:

: $\backslash operatorname\{E\}[(X-\backslash mu)^n]\; =\; s^n\backslash pi^n(2^n-2)\backslash cdot|B\_n|.$

• generalized logistic distribution
• Tukey lambda distribution
• log-logistic distribution
• half-logistic distribution
• logistic regression
• sigmoid function

{{Reflist}}

{{commons category}}

• {{Cite journal

|author1=John S. deCani |author2=Robert A. Stine

|name-list-style=amp | year = 1986

| title = A note on deriving the information matrix for a logistic distribution

| journal = The American Statistician

| volume = 40

|issue=3

| pages = 220–222

| publisher = American Statistical Association

| doi=10.2307/2684541

|jstor=2684541

}}

• {{Cite book

| first = Balakrishnan

| last = N.

| year = 1992

| title = Handbook of the Logistic Distribution

| publisher = Marcel Dekker, New York

| isbn = 0-8247-8587-8

}}

• {{cite book

| last1= Johnson|first1= N. L.|last2= Kotz|first2= S.|last3=N. |first3=Balakrishnan

| year = 1995

| title = Continuous Univariate Distributions

| others = Vol. 2

| edition = 2nd

| isbn = 0-471-58494-0 }}

• Modis, Theodore (1992) Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future, Simon & Schuster, New York. {{isbn|0-671-75917-5}}

{{ProbDistributions|continuous-infinite}}

{{Authority control}}

{{DEFAULTSORT:Logistic Distribution}}

Category:Continuous distributions

Category:Location-scale family probability distributions