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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. Image File history File links Download high resolution version (1300x975, 95 KB) Probability density function for the hyperbolic secant distribution File links The following pages link to this file: Hyperbolic secant distribution ...
Image File history File links Download high resolution version (1300x975, 72 KB) Cumulative distribution function for the Hyperbolic secant distribution File links The following pages link to this file: Hyperbolic secant distribution ...
In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...
In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...
In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds...
In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. ...
In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ...
In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...
In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ...
Entropy of a Bernoulli trial as a function of success probability. ...
Catalans constant K, which occasionally appears in estimates in combinatorics, is defined by or equivalently along with where K(x) is a complete elliptic integral of the first kind, and has nothing to do with the constant itself. ...
In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...
Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A...
Probability theory is the mathematical study of probability. ...
Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...
Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A...
In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...
A random variable follows a hyperbolic secant distribution if its probability density function (pdf) is A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...
where "sech" denotes the hyperbolic secant function. The cumulative distribution function (cdf) is In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...
where "arctan" is the inverse (circular) tangent function. The inverse cdf (or quantile function) is In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
where "arcsinh" is the inverse hyperbolic sine function and "cot" is the (circular) cotangent function. In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...
In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...
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. The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ...
In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...
In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds...
In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. ...
In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ...
In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ...
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.
- Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, Continuous Univariate Distributions, volume 2, ISBN 0417584940.
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