probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and...

use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse...

hyperbolic secant distribution, orange curve, excess kurtosis = 2 L: logistic distribution, green curve, excess kurtosis = 1.2 N: normal distribution...

include the logistic distribution, hyperbolic secant distribution, and the Champernowne distribution. The Laplace distribution is easy to integrate (if...

the hyperbolic secant function "sech", it is sometimes referred to as the sech-square(d) distribution. (See also: hyperbolic secant distribution). The...

< ∞ {\textstyle -\infty <\psi <\infty } to be the integral of the hyperbolic secant ϕ = gd ⁡ ψ ≡ ∫ 0 ψ sech ⁡ t d t = arctan ⁡ ( sinh ⁡ ψ ) . {\displaystyle...

example of a distribution that has a finite expected value but infinite variance. The hyperbolic distribution The hyperbolic secant distribution The Johnson...

In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged...

{\displaystyle \alpha ={\tfrac {\pi }{2}},y_{0}=0} ) it is the hyperbolic secant distribution. In the special case λ = 1 {\displaystyle \lambda =1} it is...

{FWHM} =2\gamma .} Another important distribution function, related to solitons in optics, is the hyperbolic secant: f ( x ) = sech ⁡ ( x X ) . {\displaystyle...

powers of secant to lower ones is fully present in this, the simplest case. The other cases are done in the same way. The utility of hyperbolic functions...

at risk that is more sensitive to the shape of the tail of the loss distribution. Expected shortfall is also called conditional value at risk (CVaR),...

the logistic distribution, the extreme value distribution, the Laplace distribution, the chi distribution, the hyperbolic secant distribution, the Wishart...

statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional...

Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the...

formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at VaR α ⁡ ( X ) {\displaystyle \operatorname...

{\displaystyle f(x)={\frac {1}{1+\left|{\frac {x-c}{a}}\right|^{2b}}}} Hyperbolic secant. This is also the derivative of the Gudermannian function. f ( x )...

^{2}/n+\mu .} The (not very famous) distribution generated by the generalized[clarification needed] hyperbolic secant distribution (NEF-GHS) has[citation needed]...

Hyper-exponential distribution Hyper-Graeco-Latin square design Hyperbolic distribution Hyperbolic secant distribution Hypergeometric distribution Hyperparameter...

In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative...

minimum. arsech – inverse hyperbolic secant function. arsinh – inverse hyperbolic sine function. artanh – inverse hyperbolic tangent function. a.s. – almost...

Proof that 22/7 exceeds π Trapezium rule Integral of the secant function Integral of secant cubed Arclength Solid of revolution Shell integration Natural...

Periodic functions Trigonometric functions: sine, cosine, tangent, cotangent, secant, cosecant, exsecant, excosecant, versine, coversine, vercosine, covercosine...

{x}{2}}\\[6pt]&={\frac {2t^{2}}{2t}}=t\\[6pt]&=\tan {\tfrac {x}{2}}\end{aligned}}} The secant integral may be evaluated in a similar manner. ∫ 0 2 π d x 2 + cos ⁡ x =...

Symmetric multivariate Laplace distribution Multivariate logistic distribution Multivariate symmetric general hyperbolic distribution In the 2-dimensional case...

Laplace distributions are extensions of the Laplace distribution and the asymmetric Laplace distribution to multiple variables. The marginal distributions of...

In statistics, Wilks' lambda distribution (named for Samuel S. Wilks), is a probability distribution used in multivariate hypothesis testing, especially...

\\[6pt]&=a^{2}\int \sec ^{3}\theta \,d\theta .\\[6pt]\end{aligned}}} The integral of secant cubed may be evaluated using integration by parts. As a result, ∫ a 2 +...

trigonometric functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of exponential functions...

the secant line, and as the two points move around the circle, the secant line varies continuously. However, when the two points collide, the secant line...