Logistic distribution

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Logistic
Probability density function
Standard logistic PDF
Cumulative distribution function
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 {{{skewness}}}
Excess kurtosis {{{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

Applications

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. ed.). ISBN 0-471-58494-0.

See also

de:Logistische Verteilung it:variabile casuale logistica


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