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In probability theory
Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
 and statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, the logistic distribution is a continuous probability distribution. Its cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 is the logistic function
Logistic function

A logistic function or logistic curve is the most common sigmoid curve. It modelsthe S-curve of growth of some set P, where P might...
, which appears in logistic regression
Logistic regression

In statistics, logistic regression is a model used for prediction of the probability of occurrence of an event by fitting data to a logistic curve....
 and feedforward neural networks. It resembles the normal distribution
Normal distribution

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields....
 in shape but has heavier tails (higher kurtosis
Kurtosis

In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real number-valued random variable....
).

>

>

Because the pdf can be expressed in terms of the square of the hyperbolic secant function
Hyperbolic function

In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric function, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", etc., in analogy to the derived trigonometric functi...
 "sech", it is sometimes referred to as the sech-square(d) distribution.

See also: hyperbolic secant distribution
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 function....


Quantile function
The inverse
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
 cumulative distribution function of the logistic distribution is , a generalization of the logit
Logit

The logit function is the inverse of the "sigmoid", or logistic function used in mathematics, especially in statistics. The logit of a number p between 0 and 1 is given by the formula:...
 function, defined as follows:
Alternative parameterization
An alternative parameterization of the logistic distribution can be derived using the substitution .






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Encyclopedia


In probability theory
Probability theory

Probability theory is the branch of mathematics concerned with analysis of Statistical randomness phenomena. The central objects of probability theory are random variables, stochastic processes, and event s: mathematical abstractions of determinism events or measured quantities that may either be single occurrences or evolve over time in an a...
 and statistics
Statistics

Statistics is a Mathematics pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It also provides tools for prediction and forecasting based on data....
, the logistic distribution is a continuous probability distribution. Its cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 is the logistic function
Logistic function

A logistic function or logistic curve is the most common sigmoid curve. It modelsthe S-curve of growth of some set P, where P might...
, which appears in logistic regression
Logistic regression

In statistics, logistic regression is a model used for prediction of the probability of occurrence of an event by fitting data to a logistic curve....
 and feedforward neural networks. It resembles the normal distribution
Normal distribution

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields....
 in shape but has heavier tails (higher kurtosis
Kurtosis

In probability theory and statistics, kurtosis is a measure of the "peakedness" of the probability distribution of a real number-valued random variable....
).

Specification


Cumulative distribution function


The logistic distribution receives its name from its cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 (cdf), which is an instance of the family of logistic functions:



Probability density function


The probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 (pdf) of the logistic distribution is given by:



Because the pdf can be expressed in terms of the square of the hyperbolic secant function
Hyperbolic function

In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric function, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", etc., in analogy to the derived trigonometric functi...
 "sech", it is sometimes referred to as the sech-square(d) distribution.

See also: hyperbolic secant distribution
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 function....


Quantile function


The inverse
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
 cumulative distribution function of the logistic distribution is , a generalization of the logit
Logit

The logit function is the inverse of the "sigmoid", or logistic function used in mathematics, especially in statistics. The logit of a number p between 0 and 1 is given by the formula:...
 function, defined as follows:

Alternative parameterization


An alternative parameterization of the logistic distribution can be derived using the substitution . This yields the following density function:

Applications


Both the United States Chess Federation
United States Chess Federation

The United States Chess Federation is a non-profit organization, the governing chess organization within the United States, and one of the federations of the F?d?ration Internationale des ?checs....
 and FIDE have switched their formulas for calculating chess ratings from the normal distribution to the logistic distribution, see Elo rating system
Elo rating system

The Elo rating system is a method for calculating the relative skill levels of players in two-player games such as chess and Go . It is named after its creator Arpad Elo , a Hungary-born United States physics professor....
.

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

? Epidemiology - to describe the spreading of epidemics

? Psychology - to describe learning

? Technology - to describe how new technologies diffuse and substitute for each other

? Market - the diffusion of new-product sales

? Energy - the diffusion and substitution of primary energy sources

Related distributions

If log(X) has a logistic distribution then X has a log-logistic distribution
Log-logistic distribution

In probability and statistics, the log-logistic distribution is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, for example mortality from cancer following diagnosis or treatment....
 and X – a has a shifted log-logistic distribution.

Derivations


Expected Value


Substitute:



Note the odd function:


See also

  • logistic regression
    Logistic regression

    In statistics, logistic regression is a model used for prediction of the probability of occurrence of an event by fitting data to a logistic curve....
  • sigmoid function
    Sigmoid function

    Many natural processes and complex system learning curve display a history dependent progression from small beginnings that accelerates and approaches a climax over time....