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logistic distribution: Information from Answers.com

️Wed Jul 01 2015

Logistic

Probability density function
Cumulative distribution function
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)	$e^{m\,t}\,\mathrm{B}(1-s\,t,\;1+s\,t)\!$ for $\|s\,t\|<1\!$
Characteristic function	$e^{imt}\,\mathrm{B}(1-ist,\;1+ist)\,$ for $\|ist\|<1\,$

In probability theory and statistics, the logistic distribution is a continuous probability distribution. It is the distribution whose cumulative distribution function is the standard logistic function.^[vague]

This distribution has longer tails than the normal distribution and a higher kurtosis of 1.2 (compared with 0 for the normal distribution).

Related to the logistic distribution is the half-logistic distribution.

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:

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

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.

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