# Logistic distribution

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

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:

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

### Probability density function

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

${\displaystyle f(x;\mu ,s)={\frac {e^{-(x-\mu )/s}}{s\left(1+e^{-(x-\mu )/s}\right)^{2}}}\!}$
${\displaystyle ={\frac {1}{4\,s}}\;\operatorname {sech} ^{2}\!\left({\frac {x-\mu }{2\,s}}\right).}$

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 ${\displaystyle F^{-1}}$, a generalization of the logit function, defined as follows:

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

## Alternative parameterization

An alternative parameterization of the logistic distribution can be derived using the substitution ${\displaystyle \sigma ^{2}=\pi ^{2}\,s^{2}/3}$. This yields the following density function:

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

## Generalized log-logistic distribution

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

Parameters Probability density function Cumulative distribution function ${\displaystyle \mu \in (-\infty ,\infty )\,}$ location (real) ${\displaystyle \sigma \in (0,\infty )\,}$ scale (real) ${\displaystyle \xi \in (-\infty ,\infty )\,}$ shape (real) ${\displaystyle x\geqslant \mu -\sigma /\xi \,\;(\xi \geqslant 0)}$ ${\displaystyle x\leqslant \mu -\sigma /\xi \,\;(\xi <0)}$ ${\displaystyle {\frac {(1+\xi z)^{-(1/\xi +1)}}{\sigma \left(1+(1+\xi z)^{-1/\xi }\right)^{2}}}}$ where ${\displaystyle z=(x-\mu )/\sigma \,}$ ${\displaystyle \left(1+(1+\xi z)^{-1/\xi }\right)^{-1}\,}$ where ${\displaystyle z=(x-\mu )/\sigma \,}$ ${\displaystyle \mu +{\frac {\sigma }{\xi }}(\alpha \csc(\alpha )-1)}$ where ${\displaystyle \alpha =\pi \xi \,}$ ${\displaystyle \mu \,}$ ${\displaystyle \mu +{\frac {\sigma }{\xi }}\left[\left({\frac {1-\xi }{1+\xi }}\right)^{\xi }-1\right]}$ ${\displaystyle {\frac {\sigma ^{2}}{\xi ^{2}}}[2\alpha \csc(2\alpha )-(\alpha \csc(\alpha ))^{2}]}$ where ${\displaystyle \alpha =\pi \xi \,}$ {{{skewness}}} {{{kurtosis}}}
${\displaystyle F_{(\xi ,\mu ,\sigma )}(x)=\left(1+\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }\right)^{-1}}$

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

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

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

## 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. ed.). ISBN 0-471-58494-0.