# logistic distribution

Probability density function | |

Cumulative distribution function | |

Parameters | location (real) scale (real) |
---|---|

Support | |

Probability density function (pdf) | |

Cumulative distribution function (cdf) | |

Mean | |

Median | |

Mode | |

Variance | |

Skewness | |

Excess kurtosis | |

Entropy | |

Moment-generating function (mgf) | for , Beta function |

Characteristic function | for |

**logistic distribution**is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks.

## Specification

### Cumulative distribution function

The logistic distribution receives its name from its cumulative distribution function (cdf), which is an instance of the family of logistic functions:- :

### Probability density function

The probability density function (pdf) of the logistic distribution is given by:- :

Because the pdf 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

### Quantile function

The inverse cumulative distribution function of the logistic distribution is , a generalization of the logit 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:## Generalized log-logistic distribution

The**Generalized log-logistic distribution (GLL)**has three parameters and .

Probability density function | |

Cumulative distribution function | |

Parameters |
location (real) scale (real) shape (real) |
---|---|

Support | |

Probability density function (pdf) | where |

Cumulative distribution function (cdf) | where |

Mean | where |

Median | |

Mode | |

Variance | where |

Skewness | |

Excess kurtosis | |

Entropy | |

Moment-generating function (mgf) | |

Characteristic function |

The cumulative distribution function is

The probability density function is

again, for

## References

- N., Balakrishnan (1992).
*Handbook of the Logistic Distribution*. Marcel Dekker, New York. ISBN 0-8247-8587-8. - Johnson, N. L., Kotz, S., Balakrishnan N. (1995).
*Continuous Univariate Distributions*, Vol. 2, 2nd Ed.. ISBN 0-471-58494-0.

## See also

**location parameter**, since its value determines the "location" of the probability distribution.

In other words, when you graph the function, the

**location parameter**determines where the origin will be located.

**.....**Click the link for more information.

In mathematics, the

**real numbers**may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and**.....**Click the link for more information. In probability theory and statistics, a

**scale parameter**is a special kind of numerical parameter of a parametric family of probability distributions.## Definition

If a family of probability densities with parameter*s*is of the form**.....**Click the link for more information. In mathematics, a

**support**of a function*f*from a set*X*to the real numbers**R**is a subset*Y*of*X*such that*f*(*x*) is zero for all*x*in*X*and outside*Y*.**.....**Click the link for more information. In mathematics, a

Formally, a probability distribution has density

**probability density function (pdf)**is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density

*f*, if*f***.....**Click the link for more information. In probability theory, the

**cumulative distribution function**(CDF), also called**probability distribution function**or just**distribution function**,^{[1]}completely describes the probability distribution of a real-valued random variable*X*.**.....**Click the link for more information.**expected value**(or

**mathematical expectation**, or

**mean**) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).

**.....**Click the link for more information.

**median**is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The

*median*of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking

**.....**Click the link for more information.

In statistics,

**mode**means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.**.....**Click the link for more information.**variance**of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.

**.....**Click the link for more information.

**skewness**is a measure of the asymmetry of the probability distribution of a real-valued random variable.

## Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.**.....**Click the link for more information.

**kurtosis**(from the Greek word

*kurtos*, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent

**.....**Click the link for more information.

**Shannon entropy**or

**information entropy**is a measure of the uncertainty associated with a random variable.

Shannon entropy quantifies the information contained in a piece of data: it is the minimum average message length, in bits (if using base-2 logarithms), that must

**.....**Click the link for more information.

In probability theory and statistics, the

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.

**moment-generating function**of a random variable*X*iswherever this expectation exists. The moment-generating function generates the moments of the probability distribution.

**.....**Click the link for more information.**beta function**, also called the Euler integral of the first kind, is a special function defined by

for Re(

*x*), Re(

*y*) > 0.

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet.

**.....**Click the link for more information.

In probability theory, the

**characteristic function**of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where*X*is any random variable with the distribution in question:**.....**Click the link for more information.**Probability theory**is the branch of mathematics concerned with analysis of random phenomena.

^{[1]}The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities

**.....**Click the link for more information.

**Statistics**is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities.

**.....**Click the link for more information.

In probability theory, the

**cumulative distribution function**(CDF), also called**probability distribution function**or just**distribution function**,^{[1]}completely describes the probability distribution of a real-valued random variable*X*.**.....**Click the link for more information.**logistic function**or

**logistic curve**models the S-curve of growth of some set

*P*. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.

**.....**Click the link for more information.

In statistics,

**logistic regression**is a regression model for binomially distributed response/dependent variables. It is useful for modeling the probability of an event occurring as a function of other factors.**.....**Click the link for more information.**feedforward neural network**is an artificial neural network where connections between the units do

*not*form a directed cycle. This is different from recurrent neural networks.

**.....**Click the link for more information.

In probability theory, the

**cumulative distribution function**(CDF), also called**probability distribution function**or just**distribution function**,^{[1]}completely describes the probability distribution of a real-valued random variable*X*.**.....**Click the link for more information. In mathematics, a

Formally, a probability distribution has density

**probability density function (pdf)**is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density

*f*, if*f***.....**Click the link for more information.**hyperbolic functions**are analogs of the ordinary trigonometric, 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.*

**.....**Click the link for more information.

**hyperbolic secant distribution**is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function.

**.....**Click the link for more information.

**inverse function**for ƒ, denoted by ƒ

^{−1}, is a function in the opposite direction, from

*B*to

*A*, with the property that a round trip (a composition) returns each element to itself.

**.....**Click the link for more information.

**logit**(pronounced with a long "o" and a soft "g", IPA /loʊdʒɪt/) of a number

*p*between 0 and 1 is

**.....**Click the link for more information.

**location parameter**, since its value determines the "location" of the probability distribution.

In other words, when you graph the function, the

**location parameter**determines where the origin will be located.

**.....**Click the link for more information.

In mathematics, the

**real numbers**may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and**.....**Click the link for more information.This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.