To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Hyperbolic secant distribution

From Wikipedia, the free encyclopedia

hyperbolic secant
Probability density function
Plot of the hyperbolic secant PDF
Cumulative distribution function
Plot of the hyperbolic secant CDF
Ex. kurtosis
Entropy4/π K
MGF for
CF for

In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution.

YouTube Encyclopedic

  • 1/5
    9 195
    128 708
    5 874
  • ✪ FMSP Revision: MEI DE Numerical Methods for Differential Equations
  • ✪ Average distance on a sphere | MIT 18.02SC Multivariable Calculus, Fall 2010
  • ✪ Slope of Tangent Line Derivative at a Point
  • ✪ Class 12 XII Maths CBSE Inverse Trigonometric Functions 3
  • ✪ Mod-01 Lec-36 Solitonic Communication



A random variable follows a hyperbolic secant distribution if its probability density function (pdf) can be related to the following standard form of density function by a location and shift transformation:

where "sech" denotes the hyperbolic secant function. The cumulative distribution function (cdf) of the standard distribution is

where "arctan" is the inverse (circular) tangent function. The inverse cdf (or quantile function) is

where "arcsinh" is the inverse hyperbolic sine function and "cot" is the (circular) cotangent function.

The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its pdf is proportional to its characteristic function. However, the hyperbolic secant distribution is leptokurtic; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution.

Johnson et al. (1995, p147) places this distribution in the context of a class of generalized forms of the logistic distribution, but use a different parameterisation of the standard distribution compared to that here. Ding (2014) shows three occurrences of the Hyperbolic secant distribution in statistical modeling and inference.

Losev (1989) has proposed to consider the hyperbolic secant as the symmetrical case for a more general asymmetric curve , which can be scaled with to become a generalized version of the hyperbolic secant distribution.


  • Baten, W. D. (1934). "The probability law for the sum of n independent variables, each subject to the law ". Bulletin of the American Mathematical Society. 40 (4): 284–290. doi:10.1090/S0002-9904-1934-05852-X.
  • Talacko, J. (1956). "Perks' distributions and their role in the theory of Wiener's stochastic variables". Trabajos de Estadistica. 7 (2): 159–174. doi:10.1007/BF03003994.
  • Devroye, Luc (1986). Non-Uniform Random Variate Generation. New York: Springer-Verlag. Section IX.7.2.
  • Smyth, G.K. (1994). "A note on modelling cross correlations: Hyperbolic secant regression" (PDF). Biometrika. 81 (2): 396–402. doi:10.1093/biomet/81.2.396.
  • Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). Continuous Univariate Distributions. 2. ISBN 978-0-471-58494-0.
  • Ding, P. (2014). "Three Occurrences of the Hyperbolic-Secant Distribution". The American Statistician. 68: 32–35. CiteSeerX doi:10.1080/00031305.2013.867902.
  • Losev, A. (1989). "A new lineshape for fitting X‐ray photoelectron peaks". Surface and Interface Analysis. 14 (12): 845–849. doi:10.1002/sia.740141207.
This page was last edited on 20 August 2019, at 10:24
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.