Logistic distribution
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| Probability density function Image:Logisticpdfunction.png | |
| Cumulative distribution function Image:Logistic cdf.png | |
| Parameters | <math>\mu\,</math> location (real) <math>s>0\,</math> scale (real) |
|---|---|
| Support | <math>x \in (-\infty; +\infty)\!</math> |
| Probability density function (pdf) | <math>\frac{e^{-(x-\mu)/s |
| Cumulative distribution function (cdf) | {{{cdf}}} |
| Mean | {{{mean}}} |
| Median | {{{median}}} |
| Mode | {{{mode}}} |
| Variance | {{{variance}}} |
| Skewness | {{{skewness}}} |
| Excess kurtosis | {{{kurtosis}}} |
| Entropy | {{{entropy}}} |
| Moment-generating function (mgf) | {{{mgf}}} |
| Characteristic function | {{{char}}} |
cdf =<math>\frac{1}{1+e^{-(x-\mu)/s}}\!</math>|
mean =<math>\mu\,</math>|
median =<math>\mu\,</math>|
mode =<math>\mu\,</math>|
variance =<math>\frac{\pi^2}{3} s^2\!</math>|
skewness =<math>0\,</math>|
kurtosis =<math>6/5\,</math>|
entropy =<math>\ln(s)+2\,</math>|
mgf =<math>e^{\mu\,t}\,\mathrm{B}(1-s\,t,\;1+s\,t)\!</math>
for <math>|s\,t|<1\!</math>, Beta function|
char =<math>e^{i \mu t}\,\mathrm{B}(1-ist,\;1+ist)\,</math>
for <math>|ist|<1\,</math>|
}} 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.
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:
- <math>F(x; \mu,s) = \frac{1}{1+e^{-(x-\mu)/s}} \!</math>
- <math>= \frac12 + \frac12 \;\operatorname{tanh}\!\left(\frac{x-\mu}{2\,s}\right).</math>
[edit] Probability density function
The probability density function (pdf) of the logistic distribution is given by:
- <math>f(x; \mu,s) = \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2} \!</math>
- <math>=\frac{1}{4\,s} \;\operatorname{sech}^2\!\left(\frac{x-\mu}{2\,s}\right).</math>
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 <math>F^{-1}</math>, a generalization of the logit function, defined as follows:
- <math>F^{-1}(p; \mu,s) = \mu + s\,\ln\left(\frac{p}{1-p}\right).</math>
[edit] Alternative parameterization
An alternative parameterization of the logistic distribution can be derived using the substitution <math>\sigma^2 = \pi^2\,s^2/3</math>. This yields the following density function:
- <math>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).</math>
[edit] Generalized log-logistic distribution
The Generalized log-logistic distribution (GLL) has three parameters <math> \mu,\sigma \,</math> and <math> \xi</math>.
| Probability density function | |
| Cumulative distribution function | |
| Parameters | <math>\mu \in (-\infty,\infty) \,</math> location (real) <math>\sigma \in (0,\infty) \,</math> scale (real) |
|---|---|
| Support | <math>x \geqslant \mu -\sigma/\xi\,\;(\xi \geqslant 0)</math> <math>x \leqslant \mu-\sigma/\xi\,\;(\xi < 0)</math> |
| Probability density function (pdf) | <math>\frac{(1+\xi z)^{-(1/\xi +1) |
| Cumulative distribution function (cdf) | {{{cdf}}} |
| Mean | {{{mean}}} |
| Median | {{{median}}} |
| Mode | {{{mode}}} |
| Variance | {{{variance}}} |
| Skewness | {{{skewness}}} |
| Excess kurtosis | {{{kurtosis}}} |
| Entropy | {{{entropy}}} |
| Moment-generating function (mgf) | {{{mgf}}} |
| Characteristic function | {{{char}}} |
where <math>z=(x-\mu)/\sigma\,</math>|
cdf =<math>\left(1+(1 + \xi z)^{-1/\xi}\right)^{-1} \,</math>
where <math>z=(x-\mu)/\sigma\,</math>|
mean =<math>\mu + \frac{\sigma}{\xi}(\alpha \csc(\alpha)-1)</math>
where <math>\alpha= \pi \xi\, </math>|
median =<math>\mu \,</math>|
mode =<math>\mu + \frac{\sigma}{\xi}\left[\left(\frac{1-\xi}{1+\xi}\right)^\xi - 1 \right] </math>|
variance =<math> \frac{\sigma^2}{\xi^2}[2\alpha \csc(2 \alpha) - (\alpha \csc(\alpha))^2] </math>
where <math>\alpha= \pi \xi\, </math>|
entropy =| mgf =| char =|
}}
The cumulative distribution function is
- <math>F_{(\xi,\mu,\sigma)}(x) = \left(1 + \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right)^{-1}</math>
for <math> 1 + \xi(x-\mu)/\sigma \geqslant 0</math>, where <math>\mu\in\mathbb R</math> is the location parameter, <math>\sigma>0 \,</math> the scale parameter and <math>\xi\in\mathbb R</math> the shape parameter. Note that some references give the "shape parameter" as <math> \kappa = - \xi \,</math>.
The probability density function is
- <math>\frac{\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-(1/\xi +1)}}
{\sigma\left[1 + \left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right]^2} . </math>
again, for <math> 1 + \xi(x-\mu)/\sigma \geqslant 0. </math>
[edit] Applications
Both the USCF and FIDE have switched their formulas for calculating chess ratings to the logistic distribution.
| Please help improve this article or section by expanding it. Further information might be found on the talk page or at requests for expansion. (January 2007) |
[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
fr:Loi logistique it:variabile casuale logistica pl:Rozkład logistyczny
