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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the logistic distribution is a
continuous probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
. Its
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
is the
logistic function A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation f(x) = \frac where The logistic function has domain the real numbers, the limit as x \to -\infty is 0, and the limit as x \to +\infty is L. ...
, which appears in
logistic regression In statistics, a logistic model (or logit model) is a statistical model that models the logit, log-odds of an event as a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regres ...
and
feedforward neural network Feedforward refers to recognition-inference architecture of neural networks. Artificial neural network architectures are based on inputs multiplied by weights to obtain outputs (inputs-to-output): feedforward. Recurrent neural networks, or neur ...
s. It resembles the
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
in shape but has heavier tails (higher
kurtosis In probability theory and statistics, kurtosis (from , ''kyrtos'' or ''kurtos'', meaning "curved, arching") refers to the degree of “tailedness” in the probability distribution of a real-valued random variable. Similar to skewness, kurtos ...
). The logistic distribution is a special case of the Tukey lambda distribution.


Specification


Cumulative distribution function

The logistic distribution receives its name from its
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
, 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 In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
. :F(x; \mu, s) = \frac = \frac12 + \frac12 \operatorname \left(\frac\right). In this equation is the
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
, and is a scale parameter proportional to the
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
.


Probability density function

The
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
is the
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
of the cumulative distribution function: : \begin f(x; \mu,s) & = \frac = \frac \\ pt& =\frac \\ pt& =\frac \operatorname^2\left(\frac\right). \end When the location parameter  is 0 and the scale parameter  is 1, then the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
of the logistic distribution is given by : \begin f(x; 0,1) & = \frac \\ pt& = \frac 1 \\ pt& = \frac 1 4 \operatorname^2 \left(\frac x 2 \right). \end 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''. (See also:
hyperbolic secant distribution In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic se ...
).


Quantile function

The inverse cumulative distribution function (
quantile function In probability and statistics, the quantile function is a function Q: ,1\mapsto \mathbb which maps some probability x \in ,1/math> of a random variable v to the value of the variable y such that P(v\leq y) = x according to its probability distr ...
) of the logistic distribution is a generalization of the
logit In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in Data transformation (statistics), data transformations. Ma ...
function. Its derivative is called the quantile density function. They are defined as follows: :Q(p;\mu,s) = \mu + s \ln\left(\frac\right). :Q'(p;s) = \frac.


Alternative parameterization

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


Applications

The logistic distribution—and the S-shaped pattern of its
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
(the
logistic function A logistic function or logistic curve is a common S-shaped curve ( sigmoid curve) with the equation f(x) = \frac where The logistic function has domain the real numbers, the limit as x \to -\infty is 0, and the limit as x \to +\infty is L. ...
) and
quantile function In probability and statistics, the quantile function is a function Q: ,1\mapsto \mathbb which maps some probability x \in ,1/math> of a random variable v to the value of the variable y such that P(v\leq y) = x according to its probability distr ...
(the
logit function In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in Data transformation (statistics), data transformations. Ma ...
)—have been extensively used in many different areas.


Logistic regression

One of the most common applications is in
logistic regression In statistics, a logistic model (or logit model) is a statistical model that models the logit, log-odds of an event as a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regres ...
, which is used for modeling categorical
dependent variable A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical functio ...
s (e.g., yes-no choices or a choice of 3 or 4 possibilities), much as standard
linear regression In statistics, linear regression is a statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ...
is used for modeling
continuous variable In mathematics and statistics, a quantitative variable (mathematics), variable may be continuous or discrete. If it can take on two real number, real values and all the values between them, the variable is continuous in that Interval (mathemati ...
s (e.g., income or population). Specifically, logistic regression models can be phrased as
latent variable In statistics, latent variables (from Latin: present participle of ) are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or measured. Such '' latent va ...
models with
error variable In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A model with exactly one explanatory variable ...
s following a logistic distribution. This phrasing is common in the theory of
discrete choice In economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such c ...
models, where the logistic distribution plays the same role in logistic regression as the
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
does in
probit regression In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from ''probability'' + ''unit''. The purpose of the model is to e ...
. Indeed, the logistic and normal distributions have a quite similar shape. However, the logistic distribution has heavier tails, which often increases the
robustness Robustness is the property of being strong and healthy in constitution. When it is transposed into a system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, ...
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 The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''μ'' or ''E''F for brevity. The Fermi level does not include the work required to re ...
) dominate processes such as electronic conduction, with some smearing induced by temperature. However the pertinent ''probability'' distribution in
Fermi–Dirac statistics Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of part ...
is actually a simple
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with pro ...
, 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.


Hydrology

In
hydrology Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and drainage basin sustainability. A practitioner of hydrology is called a hydro ...
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 In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
. The
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
, 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 In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.


Chess ratings

The
United States Chess Federation The United States Chess Federation (also known as US Chess or USCF) is the governing body for chess competition in the United States and represents the U.S. in FIDE, The World Chess Federation (FIDE). USCF administers the official national Chess ...
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 The Elo rating system is a method for calculating the relative skill levels of players in zero-sum games such as chess or esports. It is named after its creator Arpad Elo, a Hungarian-American chess master and physics professor. The Elo system wa ...
(itself based on the normal distribution).


Related distributions

* Logistic distribution mimics the sech distribution; they are different cases of the
Champernowne distribution In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a generalization of the logistic distribution that was introduced by D. ...
. * If X \sim \mathrm(\mu, s) then kX + \ell \sim \mathrm(k\mu + \ell, , k, s). * If X \sim U(0, 1) then \mu + s \cdot \text(X) \sim \mathrm(\mu, s), where \text(X)=\log X-\log(1-X) is the
logit In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in Data transformation (statistics), data transformations. Ma ...
function. * If X \sim \mathrm(\mu_X, \beta) and Y \sim \mathrm(\mu_Y, \beta) independently then X-Y \sim \mathrm(\mu_X-\mu_Y,\beta) \,. * If X and Y \sim \mathrm(\mu, \beta) then X+Y \nsim \mathrm(2 \mu,\beta) \, (The sum is ''not'' a logistic distribution). E(X+Y) = 2\mu+2\beta\gamma \neq 2\mu = E\left(\mathrm(2 \mu,\beta) \right) . * If ''X'' ~ Logistic(''μ'', ''s'') then exp(''X'') ~ LogLogistic \left( \alpha = e^\mu, \beta = \frac 1 s \right) , and exp(''X'') + ''γ'' ~ shifted log-logistic \left( \alpha = e^\mu, \beta = \frac 1 s, \gamma \right) . * If ''X'' ~ Exponential(1) then ::\mu+s\log(e^X -1) \sim \operatorname(\mu,s). * If ''X'', ''Y'' ~ Exponential(λ) independently then ::\mu+s\log\left(\frac X Y \right) \sim \operatorname(\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 \mu and \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.


Derivations


Higher-order moments

The ''n''th-order central moment can be expressed in terms of the quantile function: : \begin \operatorname X-\mu)^n& = \int_^\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 \right)\rightn \, dp. \end This integral is well-known and can be expressed in terms of
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent function ...
s: : \operatorname X-\mu)^n= s^n\pi^n(2^n-2)\cdot, B_n, .


See also

* generalized logistic distribution * Tukey lambda distribution * log-logistic distribution *
half-logistic distribution In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. That is, for :X = , Y, \! whe ...
*
logistic regression In statistics, a logistic model (or logit model) is a statistical model that models the logit, log-odds of an event as a linear function (calculus), linear combination of one or more independent variables. In regression analysis, logistic regres ...
*
sigmoid function A sigmoid function is any mathematical function whose graph of a function, graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function, which is defined by the formula :\sigma(x ...


Notes


References

* * * *Modis, Theodore (1992) ''Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future'', Simon & Schuster, New York. {{DEFAULTSORT:Logistic Distribution Continuous distributions Location-scale family probability distributions