logistic distribution

Logistic
Probability density function
Enlarge picture
Standard logistic PDF
Cumulative distribution function
Enlarge picture
Standard logistic CDF
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
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 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 .

Generalized log-logistic
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
for , where is the location parameter, the scale parameter and the shape parameter. Note that some references give the "shape parameter" as .

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

Probability distributions    [ edit] ]
Univariate Multivariate
Discrete: Benford • BernoullibinomialBoltzmanncategoricalcompound Poisson • discrete phase-type • degenerateGauss-Kuzmingeometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniformYule-SimonzetaZipfZipf-MandelbrotEwensmultinomialmultivariate Polya
Continuous: BetaBeta primeCauchychi-squareDirac delta function • Coxian • Erlangexponentialexponential powerFfading • Fermi-Dirac • Fisher's zFisher-TippettGammageneralized extreme valuegeneralized hyperbolicgeneralized inverse GaussianHalf-LogisticHotelling's T-squarehyperbolic secanthyper-exponentialhypoexponentialinverse chi-square (scaled inverse chi-square) • inverse Gaussianinverse gamma (scaled inverse gamma) • KumaraswamyLandauLaplace • Lvy • Lvy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speedNakagaminormal (Gaussian)normal-gammanormal inverse GaussianParetoPearson • phase-type • polarraised cosineRayleigh • relativistic Breit-Wigner • Riceshifted GompertzStudent's ttriangulartruncated normaltype-1 Gumbeltype-2 GumbeluniformVariance-GammaVoigtvon MisesWeibullWigner semicircleWilks' lambdaDirichletGeneralized Dirichlet distribution . inverse-WishartKentmatrix normalmultivariate normalmultivariate Studentvon Mises-FisherWigner quasiWishart
Miscellaneous: bimodalCantorconditional • equilibrium • exponential family • infinitely divisible • location-scale familymarginalmaximum entropyposterior • prior • quasisamplingsingular
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.
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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
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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


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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.
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In mathematics, a 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
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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.
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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).
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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
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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.
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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.
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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.
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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
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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
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In probability theory and statistics, the moment-generating function of a random variable X is



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


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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
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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.
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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.
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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.
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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.
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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.
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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 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.
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hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function.
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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.
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logit (pronounced with a long "o" and a soft "g", IPA /loʊdʒɪt/) of a number p between 0 and 1 is


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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.


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