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Encyclopedia > Logistic distribution
Logistic
Probability density function
Standard logistic PDF
Cumulative distribution function
Standard logistic CDF
Parameters mu, location (real)
s>0, scale (real)
Support x in (-infty; +infty)!
Probability density function (pdf) frac{e^{-(x-mu)/s}} {sleft(1+e^{-(x-mu)/s}right)^2}!
Cumulative distribution function (cdf) frac{1}{1+e^{-(x-mu)/s}}!
Mean mu,
Median mu,
Mode mu,
Variance frac{pi^2}{3} s^2!
Skewness 0,
Excess kurtosis 6/5,
Entropy ln(s)+2,
Moment-generating function (mgf) e^{mu,t},mathrm{B}(1-s,t,;1+s,t)!
for |s,t|<1!
Characteristic function e^{i mu t},mathrm{B}(1-ist,;1+ist),
for |ist|<1,

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... A graph of a normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... Logistic curve, specifically the sigmoid function A 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 competition arises, the growth slows, and at maturity, growth stops. ... Logistic regression is a statistical regression model for Bernoulli-distributed dependent variables. ... In a feed forward network information at a later level, never backpropagates to a previous level A feedforward neural network is an artificial neural network where connections between the units do not form a directed cycle. ...


The logistic distribution has longer tails than the normal distribution and a higher kurtosis of 1.2 (compared with 0 for the normal distribution). A related distribution is the half-logistic distribution. The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... 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. ...

Contents

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: In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

F(x; mu,s) = frac{1}{1+e^{-(x-mu)/s}} !
= frac12 + frac12 ;operatorname{tanh}!left(frac{x-mu}{2,s}right) !

Probability density function

The probability density function (pdf) of the logistic distribution is given by: In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

f(x; mu,s) = frac{e^{-(x-mu)/s}} {sleft(1+e^{-(x-mu)/s}right)^2} !
=frac{1}{4,s} ;operatorname{sech}^2!left(frac{x-mu}{2,s}right) !

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. In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...

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. ...

Quantile function

The inverse cumulative distribution function of the logistic distribution is F − 1, a generalization of the logit function, defined as follows: In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, especially as applied in statistics, the logit (pronounced with a long o and a soft g, IPA ) of a number p between 0 and 1 is This function is used in logistic regression. ...

F^{-1}(p; mu,s) = mu + s,lnleft(frac{p}{1-p}right) !

Alternative parameterization

An alternative parameterization of the logistic distribution can be derived using the substitution sigma^2 = pi^2,s^2/3. This yields the following density function:

g(x;mu,sigma) = f(x;mu,sigmasqrt{3}/pi) = frac{pi}{sigma,4sqrt{3}} ,operatorname{sech}^2!left(frac{pi}{2 sqrt{3}} ,frac{x-mu}{sigma}right) !

Generalized log-logistic distribution

The Generalized log-logistic distribution (GLL) has three parameters mu,sigma , and &# 0;,.

Generalized log-logistic
Probability density function
Cumulative distribution function
Parameters mu in (-infty,infty) , location (real)

sigma in (0,infty) , scale (real)
&# 0;in (-infty,infty) , shape (real) Please refer to Real vs. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In probability theory and statistics, a shape parameter is a special kind of numerical parameter of a parametric family of probability distributions. ...

Support x geqslant mu -sigma/&# 0;,;(&# 0;geqslant 0)

x leqslant mu-sigma/&# 0;,;(&# 0;< 0) In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

Probability density function (pdf) frac{(1+&# 0;z)^{-(1/&# 0;+1)}}{sigmaleft(1 + (1+&# 0;z)^{-1/&# 0;right)^2}

where z=(x-mu)/sigma, In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

Cumulative distribution function (cdf) left(1+(1 + &# 0;z)^{-1/&# 0;right)^{-1} ,

where z=(x-mu)/sigma, In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

Mean mu + frac{sigma}{&# 0;(alpha csc(alpha)-1)

where alpha= pi &# 0;, In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

Median mu ,
Mode mu + frac{sigma}{&# 0;left[left(frac{1-&# 0;{1+&# 0;right)^&# 0;- 1 right]
Variance frac{sigma^2}{&# 0;2}[2alpha csc(2 alpha) - (alpha csc(alpha))^2]

where alpha= pi &# 0;, In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

Skewness {{{skewness}}}
Excess kurtosis {{{kurtosis}}}
Entropy
Moment-generating function (mgf)
Characteristic function

The cumulative distribution function is Example of the experimental data with non-zero skewness (gravitropic response of wheat coleoptiles, 1,790) In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

F_{(&# 0;mu,sigma)}(x) = left(1 + left(1+ frac{&# 0;x-mu)}{sigma}right)^{-1/&# 0;right)^{-1}

for 1 + &# 0;x-mu)/sigma geqslant 0, where muinmathbb R is the location parameter, sigma>0 , the scale parameter and &# 0;inmathbb R the shape parameter. Note that some references give the "shape parameter" as kappa = - &# 0;,.



The probability density function is In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...

frac{left(1+frac{&# 0;x-mu)}{sigma}right)^{-(1/&# 0;+1)}} {sigmaleft[1 + left(1+frac{&# 0;x-mu)}{sigma}right)^{-1/&# 0;right]^2} .

again, for 1 + &# 0;x-mu)/sigma geqslant 0.


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

Image:Bvn-small.png Probability distributions [ view • talk • edit ]
Univariate Multivariate
Discrete: Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot Ewens • multinomial • multivariate Polya
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square)• inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular•truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • equilibrium • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

  Results from FactBites:
 
Logistic distribution - Wikipedia, the free encyclopedia (430 words)
The logistic distribution is closely related to the logistic function and the logistic equation which also follow from the work of Verhulst.
Related to the logistic distribution is the half-logistic distribution.
The inverse cumulative distribution function of the logistic distribution is F
The Logistic Distribution (526 words)
The logistic distribution has been used for growth models, and is used in a certain type of regression known as the logistic regression.
The shape of the logistic distribution is very similar to that of the normal distribution.
The main difference between the normal distribution and logistic distribution lies in the tails and in the behavior of the failure rate function.
  More results at FactBites »


 

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