Hyperbolic secant distribution
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Probability density function |
|
Cumulative distribution function |
|
| Parameters | none |
|---|---|
| Support |  |
| Probability density function (pdf) |  |
| Cumulative distribution function (cdf) | ![\frac{2}{\pi} \arctan\!\left[\exp\!\left(\frac{\pi}{2}\,x\right)\right]\!](../../../../math/b/5/2/b52d9c3605d3f67e321f06edcb099f95.png) |
| Mean | 0 |
| Median | 0 |
| Mode | 0 |
| Variance | 1 |
| Skewness | 0 |
| Excess kurtosis | 2 |
| 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.
[edit] Explanation
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
![F(x) = \frac12 + \frac{1}{\pi} \arctan\!\left[\operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\right]
\!](../../../../math/2/7/8/2780e9380ed67d203ad1151bffa11ffe.png)
![= \frac{2}{\pi} \arctan\!\left[\exp\left(\frac{\pi}{2}\,x\right)\right] \!](../../../../math/b/8/2/b82c29369306f42f1f8b384a6e50574c.png)
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] \!](../../../../math/4/e/9/4e91eeaae8be71dfa588977541765e42.png)
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
", 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". Biometrika 81: 396–402. doi:.
- Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, Continuous Univariate Distributions, volume 2, ISBN 0-471-58494-0.
