Logistic distribution

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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}} {s\left(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^{m\,t}\,\mathrm{B}(1-s\,t,\;1+s\,t)\!
for |s\,t|<1\!
Characteristic function e^{imt}\,\mathrm{B}(1-ist,\;1+ist)\,
for |ist|<1\,

In probability theory and statistics, the logistic distribution is a continuous probability distribution. It is the distribution whose cumulative distribution function is the standard logistic function.[vague]

This distribution has longer tails than the normal distribution and a higher kurtosis of 1.2 (compared with 0 for the normal distribution).

Related to the logistic distribution is the half-logistic distribution.

Contents

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

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

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

f(x; \mu,s) = \frac{e^{-(x-\mu)/s}} {s\left(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.

See also: hyperbolic secant distribution

The inverse cumulative distribution function of the logistic distribution is F − 1, a generalization of the logit function, defined as follows:

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

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

  • 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. 

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