Logistic distribution
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Template:Probability distribution In probability theory and statistics, the logistic distribution is a continuous probability distribution. It is derived from the work of Pierre François Verhulst (1804–1849), Professor of Analysis at the Belgian Military College, in modelling the growth of population in Belgium in the early 1800's. Verhulst's description of the growth of population follows the cumulative distribution function of the logistic distribution (also known as the "logistic ogive"). Population grows geometrically for small populations which have more resources than they need, then becomes constant as the resources become fully utilized, then becomes very slow as the population demand for resources exceeds the supply. The logistic distribution is closely related to the logistic function and the logistic equation which also follow from the work of Verhulst.
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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\mid m,s) = \frac{1}{1+e^{-(x-m)/s}} \!<math>
- <math>= \frac12 + \frac12 \;\operatorname{tanh}\!\left(\frac{x-m}{2\,s}\right) \!<math>
Probability density function
The probability density function (pdf) of the logistic distribution is given by:
- <math>f(x\mid m,s) = \frac{e^{-(x-m)/s}} {s\left(1+e^{-(x-m)/s}\right)^2} \!<math>
- <math>=\frac{1}{4\,s} \;\operatorname{sech}^2\!\left(\frac{x-m}{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\mid m,s) = m + 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> (as well as <math>\mu=m<math>). This yields the following density function:
- <math>g(x\mid\mu,\sigma) = f(x\mid\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>
This distribution has mean <math>\mu<math>, variance <math>\sigma^2<math> and is symmetric, like the normal distribution. However, it has longer tails than the normal distribution and a higher kurtosis of 1.2 (compared with 0 for the normal distribution).
References
- Balakrishnan, N., "Handbook of the Logistic Distribution", New York: Marcel Dekker, 1992. ISBN 0824785878
- Johnson, N. L., Kotz, S., Balakrishnan N., "Continuous Univariate Distributions", volume 2, second edition, New York, 1995. ISBN 0471584940