# logistic distribution

Parameters Probability density functionStandard logistic PDF Cumulative distribution functionStandard logistic CDF location (real) scale (real) for , Beta 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.

### 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 .

Parameters Probability density function Cumulative distribution function location (real) scale (real) shape (real) where where where where

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.

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
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In other words, when you graph the function, the location parameter determines where the origin will be located.
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## Definition

If a family of probability densities with parameter s is of the form

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Formally, a probability distribution has density f, if f
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skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

## Introduction

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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 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.
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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