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| Probability density function|
Standard logistic PDF
| Cumulative distribution function|
Standard logistic CDF
|Parameters|| location (real)|
for , Beta function
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. It resembles the normal distribution in shape but has heavier tails (higher kurtosis).
Cumulative distribution function Edit
The logistic distribution receives its name from its cumulative distribution function (cdf), which is an instance of the family of logistic functions:
Probability density function Edit
The probability density function (pdf) of the logistic distribution is given by:
- See also: hyperbolic secant distribution
Quantile function Edit
Alternative parameterization Edit
An alternative parameterization of the logistic distribution, in terms of the variance σ2, can be derived using the substitution . This yields the following density function:
The logistic distribution and the S-shaped pattern that results from it have been extensively used in many different areas, including:
- Biology – to describe how species populations grow in competition
- Epidemiology – to describe the spreading of epidemics
- Psychology – to describe learning
- Technology – to describe how new technologies diffuse and substitute for each other
- Marketing – the diffusion of new-product sales
- Energy – the diffusion and substitution of primary energy sources, as in the Hubbert curve
- Hydrology - In hydrology the distribution of long duration river discharge and rainfall (e.g. monthly and yearly totals, consisting of the sum of 30 respectively 360 daily values) is often thought to be almost normal according to the central limit theorem. The normal distribution, however, needs a numeric approximation. As the logistic distribution, which can be solved analytically, is similar to the normal distribution, it can be used instead. The blue picture illustrates an example of fitting the logistic distribution to ranked October rainfalls - that are almost normally distributed - and it shows the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.
- Physics - the cdf of this distribution describes a Fermi gas and more specifically the number of electrons within a metal that can be expected to occupy a given quantum state. Its range is between 0 and 1, reflecting the Pauli exclusion principle. The value is given as a function of the kinetic energy corresponding to that state and is parametrized by the Fermi energy and also the temperature (and Boltzmann constant). By changing the sign in front of the "1" in the denominator, one goes from Fermi–Dirac statistics to Bose–Einstein statistics. In this case, the expected number of particles (bosons) in a given state can exceed unity, which is indeed the case for systems such as lasers.
Related distributions Edit
- If then
- Logistic distribution mimics Sech distribution
- If (Uniform distribution (continuous)) then
- If (Exponential distribution) then
- If and then
- If and (Gumbel distribution) then
- If and (Generalized extreme value distribution) then
- If and then
- If then (log-logistic distribution) and (shifted log-logistic distribution)
Higher order moments Edit
The n-th order central moment can be expressed in terms of the quantile function:
See also Edit
- ↑ Johnson, Kotz & Balakrishnan (1995, p.116).
- ↑ P. F. Verhulst (1845) "Recherches mathématiques sur la loi d'accroissement de la population", Nouveaux Mémoirs de l'Académie Royale des Sciences et des Belles-Lettres de Bruxelles, vol. 18)
- ↑ Lotka, Alfred J. (1925) Elements of Physical Biology, Baltimore, MD: Williams & Wilkins Co..
- ↑ Modis (1992,pp 97-105)
- ↑ Modis (1992, Chapter 2)
- ↑ J. C. Fisher and R. H. Pry (1971) "A Simple Substitution Model of Technological Change", Technological Forecasting & Social Change, vol. 3, no. 1 Template:Pn
- ↑ Modis, Theodore (1998), Conquering Uncertainty, McGraw-Hill, New York (Chapter 1)
- ↑ Cesare Marchetti (1977) "Primary Energy Substitution Models: On the Interaction between Energy and Society", Technological Forecasting & Social Change, vol. 10, Template:Page needed.
- ↑ Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis, 175–224, Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands.
- ↑ 
- John S. deCani and Robert A. Stine (1986). A note on deriving the information matrix for a logistic distribution. American Statistician 40:220-222.
- N., Balakrishnan (1992). Handbook of the Logistic Distribution, Marcel Dekker, New York.
- Johnson, N. L., Kotz, S., Balakrishnan N. (1995). Continuous Univariate Distributions, Vol. 2, 2nd Ed..
- Modis, Theodore (1992) Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future, Simon & Schuster, New York. ISBN 0671759175
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