# Logistic distribution

Parameters Probability density function Cumulative distribution function μ location (real) s > 0 scale (real) x ∈ (−∞, ∞) $\frac{e^{-\frac{x-\mu}{s}}} {s\left(1+e^{-\frac{x-\mu}{s}}\right)^2}\!$ $\frac{1}{1+e^{-\frac{x-\mu}{s}}}\!$ μ μ μ $\tfrac{1}{3}s^2 \pi^2$ 0 6/5 ln(s)+2 $e^{\mu t}\,\mathrm{B}(1-st, 1+st)$ for st ∈ (−1, 1) Beta function $e^{it\mu}\frac{\pi st}{\sinh(\pi st)}$

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).

## Specification

### Probability density function

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

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

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.[1]

### 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:

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

In this equation, x is the random variable, μ is the mean, and s is a parameter proportional to the standard deviation.

### 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, in terms of the variance σ2, can be derived using the substitution σ2 = (s2π2)/3. This yields the following density function:

$g(x;\mu,\sigma) = f \left(x;\mu,\frac{\sigma\sqrt{3}}{\pi} \right) = \frac{1}{4} \frac{\pi}{\sigma\sqrt{3}} \operatorname{sech}^2\!\left(\frac{\pi}{2 \sqrt{3}} \,\frac{x-\mu}{\sigma}\right).$

## Applications

The logistic distribution — and the S-shaped pattern of its cumulative distribution function (the logistic function) and quantile function (the logit function) — have been extensively used in many different areas. One of the most common applications is in logistic regression, which is used for modeling categorical dependent variables (e.g. yes-no choices or a choice of 3 or 4 possibilities), much as standard linear regression is used for modeling continuous variables (e.g. income or population). Specifically, logistic regression models can be phrased as latent variable models with error variables following a logistic distribution. This phrasing is common in the theory of discrete choice models, where the logistic distribution plays the same role in logistic regression as the normal distribution does in probit regression. Indeed, the logistic and normal distributions have a quite similar shape. However, the logistic distribution has heavier tails, which often increases the robustness of analyses based on it compared with using the normal distribution.

Other applications:

Fitted cumulative logistic distribution to October rainfalls using CumFreq, see also Distribution fitting
• Biology – to describe how species populations grow in competition[2][3]
• Epidemiology – to describe the spreading of epidemics[4]
• Psychology – to describe learning[5]
• Technology – to describe how new technologies diffuse and substitute for each other[6]
• Marketing – the diffusion of new-product sales[7]
• Energy – the diffusion and substitution of primary energy sources,[8] 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.[9] 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.[citation needed] 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).[citation needed] By changing the sign in front of the "1" in the denominator,[clarification needed] 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.[citation needed]

Both the United States Chess Federation and FIDE have switched their formulas for calculating chess ratings from the normal distribution to the logistic distribution; see Elo rating system.

The logistic distribution arises as limit distribution of a finite-velocity damped random motion described by a telegraph process in which the random times between consecutive velocity changes have independent exponential distributions with linearly increasing parameters.[10]

## Related distributions

$\mu-\beta\log \left(\frac{e^{-X}}{1-e^{-X}} \right) \sim \mathrm{Logistic}(\mu,\beta).$
• If X, Y ~ Exponential(1) then
$\mu-\beta\log\left(\frac{X}{Y}\right) \sim \mathrm{Logistic}(\mu,\beta).$

## Derivations

### Higher order moments

The n-th order central moment can be expressed in terms of the quantile function:

$\operatorname{E}[(X-\mu)^n] = \int_{-\infty}^\infty (x-\mu)^n dF(x) = \int_0^1\big(F^{-1}(p)-\mu\big)^n dp = s^n \int_0^1 \left[\ln\!\left(\frac{p}{1-p}\right)\right]^n dp.$

This integral is well-known[11] and can be expressed in terms of Bernoulli numbers:

$\operatorname{E}[(X-\mu)^n] = s^n\pi^n(2^n-2)\cdot|B_n|.$

## Criticism

William Feller criticized overuse of the distribution:

The logistic distribution function

(4.10)

$F(t)=\dfrac{1}{1+e^{-\alpha t-\beta}}, \qquad \alpha>0$

may serve as a warning. An unbelievably huge literature tried to establish a transcendental "law of logistic growth"; measured in appropriate units, practically all growth processes were supposed to be represented by a function of the form (4.10) with t representing time. Lengthy tables, complete with chi-square tests, supported this thesis for human populations, for bacterial colonies, development of railroads, etc. Both height and weight of plants and animals were found to follow the logistic law even though it is theoretically clear that these two variables cannot be subject to the same distribution. Laboratory experiments on bacteria showed that not even systematic disturbances can produce other results. Population theory relied on logistic extrapolations (even though they were demonstrably unreliable). The only trouble with the theory is that not only the logistic distribution but also the normal, the Cauchy, and other distributions can be fitted to the same material with the same or better goodness of fit. In this competition the logistic distribution plays no distinguished role whatever; most contradictory theoretical models can be supported by the same observational material.[12]

## Notes

1. ^ Johnson, Kotz & Balakrishnan (1995, p.116).
2. ^ 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)
3. ^ Lotka, Alfred J. (1925) Elements of Physical Biology, Baltimore, MD: Williams & Wilkins Co..
4. ^ Modis (1992,pp 97-105)
5. ^ Modis (1992, Chapter 2)
6. ^ J. C. Fisher and R. H. Pry (1971) "A Simple Substitution Model of Technological Change", Technological Forecasting & Social Change, vol. 3, no. 1[page needed]
7. ^ Modis, Theodore (1998), Conquering Uncertainty, McGraw-Hill, New York (Chapter 1)
8. ^ Cesare Marchetti (1977) "Primary Energy Substitution Models: On the Interaction between Energy and Society", Technological Forecasting & Social Change, vol. 10,[page needed].
9. ^ Ritzema (ed.), H.P. (1994). Frequency and Regression Analysis. Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224. ISBN 90-70754-33-9.
10. ^ A. Di Crescenzo, B. Martinucci (2010) "A damped telegraph random process with logistic stationary distribution", J. Appl. Prob., vol. 47, p. 84-96.
11. ^
12. ^ William Feller, An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. (New York: John Wiley & Sons, 1971), 52-53.

## References

• John S. deCani and Robert A. Stine (1986). "A note on deriving the information matrix for a logistic distribution". The American Statistician (American Statistical Association) 40: 220–222.
• 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.
• Modis, Theodore (1992) Predictions: Society's Telltale Signature Reveals the Past and Forecasts the Future, Simon & Schuster, New York. ISBN 0-671-75917-5