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
Support x \in (-\infty; +\infty)\!
Probability density function (pdf) \frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!
Cumulative distribution function (cdf) \frac{2}{\pi} \arctan\!\left[\exp\!\left(\frac{\pi}{2}\,x\right)\right]\!
Mean 0
Median 0
Mode 0
Variance 1
Skewness 0
Excess Kurtosis 2
Entropy 4/π K \;\approx 1.16624
mgf \sec(t)\! for |t|<\frac{\pi}2\!
Char. func. \operatorname{sech}(t)\! for |t|<\frac{\pi}2\!

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.

[edit] Explanation

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

f(x) = \frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!

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

F(x) = \frac12 + \frac{1}{\pi} \arctan\!\left[\operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\right] \!
= \frac{2}{\pi} \arctan\!\left[\exp\left(\frac{\pi}{2}\,x\right)\right] \!

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

F^{-1}(p) = -\frac{2}{\pi}\, \operatorname{arcsinh}\!\left[\cot(\pi\,p)\right] \!

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.

[edit] References

  • W. D. Baten, 1934, "The probability law for the sum of n independent variables, each subject to the law (2h)^{-1} \operatorname{sech}(\pi x/2h)", 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.
  • Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, Continuous Univariate Distributions, volume 2, ISBN 0-471-58494-0.
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