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^{\mu\,t}\,\mathrm{B}(1-s\,t,\;1+s\,t)\!
for |s\,t|<1\!, Beta function
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. It resembles the normal distribution in shape but has heavier tails (higher kurtosis).

Contents

[edit] Specification

[edit] 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:

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

[edit] Probability density function

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

[edit] Quantile function

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).

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


[edit] Applications

Both the United States Chess Federation and FIDE have switched their formulas for calculating chess ratings to the logistic distribution, see ELO rating system.

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[edit] Related distributions

If log(X) has a logistic distribution then X has a log-logistic distribution and X – a has a shifted log-logistic distribution.

[edit] 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. 

[edit] See also

v • d • e
Probability distributions
 
Continuous univariate supported on the whole real line