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A random variable follows a hyperbolic secant distribution if its probability density function can be related to the following standard form of density function by a location and shift transformation: = , where "sech" denotes the hyperbolic secant function.
where n is non-negative integer, Q is the Gaussian Q-function, and I is the modified Bessel function of first kind with half-integer order. The modified Bessel function of first kind with half-integer order in itself can be represented as a finite sum in terms of hyperbolic functions. In particular, for k = 1, we have
The Johnson's S U-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. [1] [2] Johnson proposed it as a transformation of the normal distribution: [1] = + ()
Since cosh x + sinh x = e x, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all integers n, ( + ) = + . If n is a rational number (but not necessarily an integer), then cosh nx + sinh nx will be one of the values of (cosh x + sinh x) n. [4]
sinh x is half the difference of e x and e −x cosh x is the average of e x and e −x. In terms of the exponential function: [1] [4] Hyperbolic sine: the odd part of the exponential function, that is, = = =.
For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function. However, in some cases, the formulas of § Definitions in terms of logarithms do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected.
In probability theory and statistics, the raised cosine distribution is a continuous probability distribution supported on the interval [, +].The probability density function (PDF) is
When "E" is used to denote "expected value", authors use a variety of stylizations: the expectation operator can be stylized as E (upright), E (italic), or (in blackboard bold), while a variety of bracket notations (such as E(X), E[X], and EX) are all used. Another popular notation is μ X.