Logistic distribution

Logistic

Probability density function Standard logistic PDF
Cumulative distribution function Standard logistic CDF
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}}}

{s\left(1+e^{-(x-\mu)/s}\right)^2}\!</math>|

 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.

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:

<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>

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

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>

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>

Generalized log-logistic distribution

The Generalized log-logistic distribution (GLL) has three parameters <math> \mu,\sigma \,</math> and <math> \xi</math>.

Generalized log-logistic

Probability density function
Cumulative distribution function
Parameters	<math>\mu \in (-\infty,\infty) \,</math> location (real) <math>\sigma \in (0,\infty) \,</math> scale (real) <math>\xi\in (-\infty,\infty) \,</math> shape (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}}}

{\sigma\left(1 + (1+\xi z)^{-1/\xi}\right)^2} </math>

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>

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

• logistic regression
• sigmoid function

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