# Logistic distribution

 Parameters Probability density functionStandard logistic PDF Cumulative distribution functionStandard logistic CDF $\mu\,$ location (real)$s>0\,$ scale (real) $x \in (-\infty; +\infty)\!$ $\frac{e^{-(x-\mu)/s {{{cdf}}} {{{mean}}} {{{median}}} {{{mode}}} {{{variance}}} {{{skewness}}} {{{kurtosis}}} {{{entropy}}} {{{mgf}}} {{{char}}} {s\left(1+e^{-(x-\mu)/s}\right)^2}\!$|
 cdf        =$\frac{1}{1+e^{-(x-\mu)/s}}\!$|
mean       =$\mu\,$|
median     =$\mu\,$|
mode       =$\mu\,$|
variance   =$\frac{\pi^2}{3} s^2\!$|
skewness   =$0\,$|
kurtosis   =$6/5\,$|
entropy    =$\ln(s)+2\,$|
mgf        =$e^{\mu\,t}\,\mathrm{B}(1-s\,t,\;1+s\,t)\!$for $|s\,t|<1\!$, Beta function|
char       =$e^{i \mu t}\,\mathrm{B}(1-ist,\;1+ist)\,$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.

## 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:

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

### 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.

### Quantile function

The inverse cumulative distribution function of the logistic distribution is $F^{-1}$, a generalization of the logit function, defined as follows:

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

## 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:

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

## Generalized log-logistic distribution

The Generalized log-logistic distribution (GLL) has three parameters $\mu,\sigma \,$ and $\xi$.

 Parameters Probability density function Cumulative distribution function $\mu \in (-\infty,\infty) \,$ location (real) $\sigma \in (0,\infty) \,$ scale (real) $\xi\in (-\infty,\infty) \,$ shape (real) $x \geqslant \mu -\sigma/\xi\,\;(\xi \geqslant 0)$ $x \leqslant \mu-\sigma/\xi\,\;(\xi < 0)$ $\frac{(1+\xi z)^{-(1/\xi +1) {{{cdf}}} {{{mean}}} {{{median}}} {{{mode}}} {{{variance}}} {{{skewness}}} {{{kurtosis}}} {{{entropy}}} {{{mgf}}} {{{char}}} {\sigma\left(1 + (1+\xi z)^{-1/\xi}\right)^2}$

where $z=(x-\mu)/\sigma\,$|

 cdf        =$\left(1+(1 + \xi z)^{-1/\xi}\right)^{-1} \,$


where $z=(x-\mu)/\sigma\,$|

 mean       =$\mu + \frac{\sigma}{\xi}(\alpha \csc(\alpha)-1)$


where $\alpha= \pi \xi\,$|

 median     =$\mu \,$|
mode       =$\mu + \frac{\sigma}{\xi}\left[\left(\frac{1-\xi}{1+\xi}\right)^\xi - 1 \right]$|
variance   =$\frac{\sigma^2}{\xi^2}[2\alpha \csc(2 \alpha) - (\alpha \csc(\alpha))^2]$


where $\alpha= \pi \xi\,$|

 entropy    =|
mgf        =|
char       =|


}}

$F_{(\xi,\mu,\sigma)}(x) = \left(1 + \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right)^{-1}$

for $1 + \xi(x-\mu)/\sigma \geqslant 0$, where $\mu\in\mathbb R$ is the location parameter, $\sigma>0 \,$ the scale parameter and $\xi\in\mathbb R$ the shape parameter. Note that some references give the "shape parameter" as $\kappa = - \xi \,$.

$\frac{\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-(1/\xi +1)}} {\sigma\left[1 + \left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right]^2} .$

again, for $1 + \xi(x-\mu)/\sigma \geqslant 0.$

## 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.