In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the hyperbolic secant distribution is a continuous probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 whose probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 and characteristic function

Characteristic function (probability theory)

In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...

 are proportional to the hyperbolic secant function

Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. The basic hyperbolic functions are the hyperbolic sine "sinh" , and the hyperbolic cosine "cosh" , from which are derived the hyperbolic tangent "tanh" and so on.Just as the points form a...

.

Explanation

A random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

 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

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 (cdf) of the standard distribution is


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


where "arsinh" is the inverse hyperbolic sine function

Inverse hyperbolic function

The inverses of the hyperbolic functions are the area hyperbolic functions. The names hint at the fact that they give the area of a sector of the unit hyperbola in the same way that the inverse trigonometric functions give the arc length of a sector on the unit circle...

 and "cot" is the (circular) cotangent function

Trigonometric function

In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

.

The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

 and zero mean

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

, median

Median

In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

 and mode

Mode (statistics)

In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....

, 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) place this distribution in the context of a class of generalised forms of the logistic distribution, but use a different parameterisation of the standard distribution compared to that here.