Hyperbolic secant distribution

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hyperbolic secant
Probability density function
Plot of the hyperbolic secant PDF
Cumulative distribution function
Plot of the hyperbolic secant CDF
Parameters none
Probability density function (pdf)
Cumulative distribution function (cdf)
Excess kurtosis
Entropy 4/π K
Moment-generating function (mgf) for
Characteristic function 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.


A random variable follows a hyperbolic secant distribution if its probability density function (pdf) is

where "sech" denotes the hyperbolic secant function. The cumulative distribution function (cdf) 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, compared with the standard normal distribution.


  • W. D. Baten, 1934, "The probability law for the sum of n independent variables, each subject to the law ", Bulletin of the American Mathematical Society 40: 284–290.
  • J. Talacko, 1956, "Perks' distributions and their role in the theory of Wiener's stochastic variables", Trabajos de Estadistica 7:159–174.
  • Luc Devroye, 1986, Non-Uniform Random Variate Generation, Springer-Verlag, New York. Section IX.7.2.
  • G.K. Smyth (1994). "A note on modelling cross correlations: Hyperbolic secant regression" (PDF). Biometrika. 81: 396–402. doi:10.1093/biomet/81.2.396.
  • Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, Continuous Univariate Distributions, volume 2, ISBN 0-471-58494-0.